*default host=cvsup.FreeBSD.org
*default base=/usr
*default prefix=/usr
*default release=cvs
*default delete use-rel-suffix
*default compress
src-all tag=.
ports-all tag=.
doc-all tag=.
cvsroot-all tag=.

cvsup -g -L 2 configuration_file

cvsup configuration_file

/usr/src/sys/i386/conf/

options SC_DISABLE_REBOOT;
options SC_HISTORY_SIZE=2000;
options SC_MOUSE_CHAR=0x3;
options MAXCONS=5;
options SC_TWOBUTTON_MOUSE

cd /usr/src/sys/i386/conf/ && config VL

cd ../compile/VL && make depend && make && make install

unload kernel
load /boot/kernel.old/kernel
boot

rm –fr /boot/kernel
cp –fr /boot/kernel.old/kernel /boot/kernel

cp –fr /boot/kernel /boot/kernel.GENERIC

-r-xr-xr-x 1 root wheel 5934584 Feb 8 02:55 /boot/kernel.GENERIC/kernel
-r-xr-xr-x 1 root wheel 3349856 Feb 14 00:10 /boot/kernel/kernel

named_enable="YES"

zone "academicdirect.ro"
	{
		type master;
		file "academicdirect.ro";
	};

$TTL 3600
academicdirect.ro. IN SOA ns.academicdirect.ro. root.academicdirect.ro. (
	2004020902; Serial
	3600; Refresh
	1800; Retry
	604800; Expire
	86400; Minimum TTL
	)
@ IN NS ns.academicdirect.ro. ; DNS Server
@ IN NS hercule.utcluj.ro. ; DNS Server
localhost IN A 127.0.0.1; Machine Name
ns IN A 193.226.7.211; Machine Name
mail IN A 193.226.7.211; Machine Name
@ IN A 193.226.7.211; Machine Name

cd /etc/namedb && sh make-localhost

ftp stream tcp4 nowait root /usr/libexec/ftpd ftpd –l # ftp IPv4 service
ftp stream tcp6 nowait root /usr/libexec/ftpd ftpd –l # ftp IPv6 service
ssh stream tcp4 nowait root /usr/sbin/sshd sshd -i -4 # ssh IPv4 service
ssh stream tcp6 nowait root /usr/sbin/sshd sshd -i -6 # ssh IPv6 service
finger stream tcp4 nowait/3/10 nobody /usr/libexec/fingerd fingerd –s # finger IPv4
finger stream tcp6 nowait/3/10 nobody /usr/libexec/fingerd fingerd –s # finger IPv6
ntalk dgram udp wait tty:tty /usr/libexec/ntalkd ntalkd # talk
pop3 stream tcp4 nowait root /usr/local/libexec/popper popper # pop3 IPv4 service
pop3 stream tcp6 nowait root /usr/local/libexec/popper popper # pop3 IPv6 service

-r-xr-xr-x 1 root wheel 60 Feb 12 12:34 /usr/local/etc/rc.d/ftpd.sh (ls –al)
/usr/libexec/ftpd -46Dh && echo -n ' ftpd' (ftpd.sh file content)

Listen 80 # httpd port
<IfModule mod_php5.c>
	AddType application/x-httpd-php .php
	AddType application/x-httpd-php-source .phps
</IfModule> # not included by the default but required to work
ServerName vl.academicdirect.ro:80

precision = 14 ; Number of significant digits displayed in floating point numbers
expose_php = On ; PHP may expose the fact that it is installed on the server
max_execution_time = 3000 ; Maximum execution time of each script, in seconds
max_input_time = 600 ; Maximum amount of time for parsing request data
memory_limit = 128M ; Maximum amount of memory a script may consume
post_max_size = 8M ; Maximum size of POST data that PHP will accept
file_uploads = On ; Whether to allow HTTP file uploads
upload_max_filesize = 8M ; Maximum allowed size for uploaded files
display_errors = On ; For production web sites, turn this feature Off

http_port 3128 # The socket addresses where Squid will listen for client requests
auth_param basic children 5
auth_param basic realm Squid proxy-caching web server
auth_param basic credentialsttl 2 hours
read_timeout 60 minutes # The read_timeout is applied on server-side connections
acl all src 0.0.0.0/0.0.0.0
acl localhost src 127.0.0.1/255.255.255.255
acl to_localhost dst 127.0.0.0/8
acl SSL_ports port 443 563
acl Safe_ports port 80 # http
acl Safe_ports port 21 # ftp
acl Safe_ports port 1025-65535 # unregistered ports
acl Safe_ports port 591 # filemaker
acl Safe_ports port 777 # multiling http
acl CONNECT method CONNECT
acl network src 172.27.211.1 172.27.211.2 193.226.7.200 192.168.211.2
http_access allow network acl ppp src 192.168.211.0/24
http_access allow ppp
http_access deny all 

$b = preg_split("/[\n]/",$a,-1,PREG_SPLIT_NO_EMPTY);//require pcre module
// split string into an array using a perl-style regular expression as a delimiter
$c = explode(" ",$b[$i]);
// splits a string on string separator and return array of components 
$c = str_replace("<","(",$c);
//replaces all occurrences of first from last with second 
$a=`uptime`; //PHP supports one execution operator: backticks (``)

$a=`cat /etc/fstab | grep swap`; // swap information 
$a=`uptime`; // show how long system has been running 
$a =`top -d2 -u -t -I -b`; // display information about the top CPU processes 
$a =`dmesg`; // display the system message buffer 
$a=`netstat -m`; // show network status 
$aa =`df -ghi`; // display free disk space 
$s=`netstat -i`; // show network status 
$s=`pkg_info`; // a utility for displaying information on software packages

define("l_max_cycle",16);
class m_c{
	var $a;//number of atoms var 
	$b;//number of bonds var 
	$c;//molecule structure var 
	$e;//seed var 
	$f;//forcefield var 
	$m;//molecule number var 
	$n;//file name var 
	$s;//sys var 
	$t;//file type var 
	$v;//view var 
	$y;//cycles structure var 
	$z;//file size
	function m_c(){ 
		$this->a=0; 
		$this->b=0;
		for($i=0;$i<l_max_cycle+1;$i++) 
			$this->y[$i][0]=0;
	}
} 
$m = new m_c; 
$m->n = $_FILES['file']['name']; 
$m->z = $_FILES['file']['size']; 
$m->t = $_FILES['file']['type']; 
$file = ""; 
$fp = fopen($_FILES['file']['tmp_name'], "rb");
while(!feof($fp)) $file .= fread($fp, 1024);
fclose($fp);
unset($fp);

function recv(&$tvv,&$t_v,$pz,&$mol){ 
	$ciclu=0;
	for($i=0;$i<$t_v[$tvv[$pz]][1];$i++){
		if(este_v($t_v[$tvv[$pz]][$i+2],$tvv,$pz)==1){ 
			$tvv[$pz+1]=$t_v[$tvv[$pz]][$i+2];
			if($pz<l_max_cycle-1)
				recv($tvv,$t_v,$pz+1,$mol);
		}
		if($t_v[$tvv[$pz]][$i+2]==$tvv[0])
			$ciclu=1;
	}
	if(($ciclu==1)&&($pz>1))
		af($tvv,$pz,$mol);
}

echo(getenv("HTTP_HOST")."\r\n");
// get the HTTP_HOST environment variable
echo(date("F j, Y, g:i a")."\r\n");
// format a local time/date
echo(microtime()."\r\n");
// the current time in seconds and microseconds 
$m->f = implode(" ", $filerc[$i]);
// joins array elements and return one string

header("Content-type: application/octet-stream");
header('Content-Disposition: attachment; filename="'. l_max_cycle.'".txt"');

<form method='post' action='hin.php' enctype='multipart/form-data'> <input type='file' name='file'> <input type='submit'> </form>

options CONSPEED=115200

ttyd0 "/usr/local/sbin/mgetty -s 115200" dialup on secure

direct NO
blocking NO
port-owner uucp
port-group uucp
port-mode 0660
toggle-dtr YES
toggle-dtr-waittime 500
data-only NO
fax-only NO
modem-type auto
init-chat "" ATS0=0Q0&D3&C1 OK
modem-check-time 3600
rings 3
answer-chat "" ATA CONNECT \c \r
answer-chat-timeout 30
autobauding NO
ringback NO
ringback-time 30
ignore-carrier false
issue-file /etc/issue
prompt-waittime 500
login-prompt @!login:
login-time 3600
diskspace 102400
notify lori
fax-owner uucp
fax-group modem
fax-mode 0660

acl ppp src 192.168.211.0/24

ifconfig_dc0_alias0="inet 192.168.211.1 netmask 255.255.255.0"

gateway_enable="YES"

if_tun_load="NO"
if_ppp_load="NO"

/AutoPPP/ - - /etc/ppp/ppp-dial

-rwxr-xr-x 1 root wheel 45 Feb 3 15:19 /etc/ppp/ppp-dial

chmod +x /etc/ppp/ppp-dial

#!/bin/sh exec /usr/sbin/ppp –direct server

server:
 set dial "ABORT BUSY ABORT NO\\sCARRIER TIMEOUT 5 \"\" AT \
 OK-AT-OK ATE1Q0 OK \\dATDT\\T TIMEOUT 40 CONNECT"
 set ifaddr 192.168.211.1 192.168.211.2-192.168.211.3
 enable pap proxy passwdauth
 accept dns

lori lori * lori *

chmod 0400 /etc/ppp/ppp.secret 

/AutoPPP/ - - /etc/ppp/pppd-dial

-rwxr-xr-x 1 root wheel 171 Feb 3 18:12 /etc/ppp/pppd-dial

chmod +x /etc/ppp/pppd-dial

#!/bin/sh exec /usr/sbin/pppd auth 192.168.211.1:192.168.211.2 192.168.211.1:192.168.211.3
	nodefaultroute ms-dns 193.226.7.211 ms-wins 193.226.7.211 ms-wins 172.27.211.2

lori academicdirect.ro lori *

chmod 0400 /etc/ppp/chap-secrets
chmod 0400 /etc/ppp/pap-secrets 

Feb 3 17:54:34 ns ppp[647]: Phase: Using interface: tun0
Feb 3 17:54:34 ns ppp[647]: Phase: deflink: Created in closed state
Feb 3 17:54:34 ns ppp[647]: tun0: Command:
	server: set dial ABORT BUSY ABORT NO\sCARRIER TIMEOUT 5 ""
	AT OK-AT-OK ATE1Q0 OK \dATDT\T TIMEOUT 40 CONNECT
Feb 3 17:54:34 ns ppp[647]: tun0: Command:
	server: set ifaddr 192.168.211.1 192.168.211.2-192.168.211.3
Feb 3 17:54:34 ns ppp[647]: tun0: Command:
	server: enable pap proxy passwdauth
Feb 3 17:54:34 ns ppp[647]: tun0: Command:
	server: accept dns
Feb 3 17:54:34 ns ppp[647]: tun0: Phase:
	PPP Started (direct mode).
Feb 3 17:54:34 ns ppp[647]: tun0: Phase:
	bundle: Establish
Feb 3 17:54:34 ns ppp[647]: tun0: Phase:
	deflink: closed -> opening
Feb 3 17:54:34 ns ppp[647]: tun0: Phase:
	deflink: Connected!
Feb 3 17:54:34 ns ppp[647]: tun0: Phase:
	deflink: opening -> carrier
Feb 3 17:54:35 ns ppp[647]: tun0: Phase:
	deflink: /dev/ttyd0: CD detected
Feb 3 17:54:35 ns ppp[647]: tun0: Phase:
	deflink: carrier -> lcp
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	bundle: Authenticate
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	deflink: his = none, mine = PAP
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	Pap Input: REQUEST (lori)
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	Pap Output: SUCCESS
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	deflink: lcp -> open
Feb 3 17:54:39 ns ppp[647]: tun0: Phase:
	bundle: Network
Feb 3 17:54:50 ns ppp[647]: tun0: Phase:
	deflink: open -> lcp
Feb 3 17:54:50 ns ppp[647]: tun0: Phase:
	bundle: Terminate
Feb 3 17:54:52 ns ppp[647]: tun0: Phase:
	deflink: Carrier lost
Feb 3 17:54:52 ns ppp[647]: tun0: Phase:
	deflink: Disconnected!
Feb 3 17:54:52 ns ppp[647]: tun0: Phase:
	deflink: Connect time: 18 secs: 2177 octets in, 14535 octets out
Feb 3 17:54:52 ns ppp[647]: tun0: Phase:
	deflink: 69 packets in, 47 packets out
Feb 3 17:54:52 ns ppp[647]: tun0: Phase:
	total 928 bytes/sec, peak 3171 bytes/sec on Tue Feb 3 17:54:48 2004
Feb 3 17:54:52 ns ppp[655]: tun0: Phase:
	deflink: lcp -> closed
Feb 3 17:54:52 ns ppp[655]: tun0: Phase:
	bundle: Dead
Feb 3 17:54:52 ns ppp[655]: tun0: Phase:
	PPP Terminated (normal).

Feb 3 17:56:47 ns pppd[656]: pppd 2.3.5 started by root, uid 0
Feb 3 17:56:47 ns pppd[656]: Using interface ppp0
Feb 3 17:56:47 ns pppd[656]: Connect: ppp0 <--> /dev/ttyd0
Feb 3 17:56:50 ns pppd[656]: sent [CHAP Challenge id=0x1
	<1e9195c5294cb50b80401cd9e34f15a89b2b823509 aef6fead7549f8b0ae5695a7710a966996>,
	name = "ns.academicdirect.ro"]
Feb 3 17:56:50 ns pppd[656]: rcvd [CHAP Response id=0x1
	<8126c29e65530d6114658931eaa57390>, name ="lori"]
Feb 3 17:56:50 ns pppd[656]: sent [CHAP Success id=0x1 "Welcome to ns.academicdirect.ro."]
Feb 3 17:56:50 ns pppd[656]: sent [IPCP ConfReq id=0x1
	<addr 192.168.211.1> <compress VJ 0f 01>]
Feb 3 17:56:50 ns pppd[656]: CHAP peer authentication succeeded for lori
Feb 3 17:56:50 ns pppd[656]: rcvd [IPCP ConfReq id=0x1 <compress VJ 0f 01>
	<addr 192.168.211.3> <ms-dns 0.0.0.0> <ms-wins 0.0.0.0> <ms-dns 0.0.0.0> <ms-wins 0.0.0.0>]
Feb 3 17:56:50 ns pppd[656]: sent [IPCP ConfNak id=0x1 <ms-dns 193.226.7.211>
	<ms-wins 193.226.7.211> <ms-dns 193.226.7.211> <ms-wins 172.27.211.2>]
Feb 3 17:56:50 ns pppd[656]: rcvd [IPCP ConfAck id=0x1
	<addr 192.168.211.1> <compress VJ 0f 01>]
Feb 3 17:56:51 ns pppd[656]: rcvd [IPCP ConfReq id=0x2 <compress VJ 0f 01>
	<addr 192.168.211.3> <ms-dns 193.226.7.211> <ms-wins 193.226.7.211>
	<ms-dns 193.226.7.211> <ms-wins 172.27.211.2>]
Feb 3 17:56:51 ns pppd[656]: sent [IPCP ConfAck id=0x2 <compress VJ 0f 01>
	<addr 192.168.211.3> <ms-dns 193.226.7.211> <ms-wins 193.226.7.211>
	<ms-dns 193.226.7.211> <ms-wins 172.27.211.2>]
Feb 3 17:56:51 ns pppd[656]: local IP address 192.168.211.1
Feb 3 17:56:51 ns pppd[656]: remote IP address 192.168.211.3
Feb 3 17:56:51 ns pppd[656]: Compression disabled by peer.
Feb 3 17:57:04 ns pppd[656]: Hangup (SIGHUP)
Feb 3 17:57:04 ns pppd[656]: Modem hangup, connected for 1 minutes
Feb 3 17:57:04 ns pppd[656]: Connection terminated, connected for 1 minutes
Feb 3 17:57:05 ns pppd[656]: Exit.

Simulation and Computer Modelling of Carbonate Concentration in Brewery Effluent

O. D. ADENIYI
Department of Chemical Engineering, Federal University of Technology, Minna, Nigeria
Email: lekanadeniyi2002@yahoo.co.uk


Abstract
The development of a mathematical model to predict the concentration of carbonates in effluent discharged from a brewery industry is the aim of this paper. This was achieved by obtaining effluent data for several years and using the method of least squares to develop the model. A mean deviation of 9% was observed by comparing the experimental data with the simulated results. The constituent parameter with the greatest influence on the simulated model was found to be sodium ion (Na⁺) with a coefficient of 0.87642 while that with the least effect was the temperature with a coefficient of 0.0514255. In addition, a control model was developed to monitor the conversions of the effluent constituents in three Continuous Stirred Tank Reactors (CSTRs), some deviation was observed between the set-point values and the empirical values.
Keywords
Effluent, BOD, pH, brewery, ions, carbonate, model, simulation


Introduction

Recently, much attention has been focused on the development of accurate equations of state for the prediction of several process parameters. Much effort has also been applied to the development of several equilibrium calculation algorithms for handling some numerical complexities that are inherent in the modeling of waste systems. Computing power, data acquisition, simulation, optimization and information systems have greatly improved effluent management over the recent years. To maintain set discharge standards, to the environment, it is imperative to adopt the most efficient effluent monitoring and management system available. In general, the optimization does not represent a radical change in operating procedures to maintain a safe discharge standard. Effluent monitoring systems provide accurate cause – and – effect relationships of a process. Obtained set point data are used by engineers as analytical tools to understand and improve the process (Luyben, 1995; Richardson and Peacock, 1994; Meyer, 1992a,b,c; Austin, 1984). In Nigeria, the last two decades have marked the emergence of several indigenous and foreign breweries. The high demand for brewery products, as well as the technological advancement in this regards, have further accelerated the growth of this industries. In Nigeria, there are nineteen breweries, and the waste generated is one of the major sources of industrial waste hazards. Improper handling and disposal could easily constitute a problem to both the people and the environment. The solid waste could be a source and reservoir to epidemic disease such as cholera and typhoid fever. The liquid effluent, though less noticeable as a waste, constitutes more hazard than the easily noticeable solid waste. The liquid effluent, on percolation into shallow wells could be a direct source of contamination to portable water. When this reached into streams, the danger to public health is as a result of the consumption of polluted water by the unsuspecting public, the aquatic as well as the land animals. Polluted water could lead to the migration of fishes from streams and rivers or outright death of the fishes. The surviving ones may become sick and unpalatable, the result of which may have an economic impact on the surrounding communities. Even when aquatic life survives in the polluted stream, it is not a proof of pollution free environment as some aquatic animals such as shell-fish have been noticed to survive and even thrive in polluted waters. Some chemical constituents of brewery waste (e.g. Chromium) could be accumulated by plants and from there, enter the food chain and be consumed by humans, this could enhance the development of some terminal disease such as cancer (Eckenfelder, 1989; Nikoladze et-al, 1989; Welson and Wenetow, 1982; Suess, 1982; Kriton, 1980). For brewery effluent, the prevailing pH value is the resultant of the disassociation of organic and inorganic compounds and their subsequent parameters are in principle targeted at measuring characteristics that could evaluate the extent or possibility of disassociation of organic or inorganic compounds and their subsequent hydrolysis. In other words measured but are not additive functions. Effluents form breweries can be characterized based on the relative oxygen demand, expressed as biochemical oxygen demand (BOD) or chemical oxygen demand COD, suspended solids (SS), pH, Temperature and flow parameters (Odigure and Adeniyi, 2002; Luyben, 1995; Imre 1984, Suess 1982). The aim of this paper is to develop a model equation to predict the concentration of carbonate ions (CO₃^2-) in brewery effluent, this would be achieved by utilizing, computer simulation techniques. In addition the paper seeks to monitor and control the concentration of CO₃^2- in 3 CSTRs connected in series.

Conceptualization and Model Development

In the effluent stream, the components of consideration are
Cl^-, CO₃^2-, N_a⁺, Ca²⁺
Na⁺ + Cl^- → N_aCl (1)
Ca²⁺ + CO₃^2- → C_aCO₃ (2)
The liquid effluent from the brewery is charged into a series of 3 CSTR, where product B is produced and reaction A is consumed according to the first order reaction, occurring in the fluid. The feed back controller used for this system, is a Proportional and Integral controller (PI). The controller looks at the product concentration leaving the third tank (C_A3) and makes adjustments in the inlet concentration to the first reaction C_A0 in order to keep C_A3 near its desired set point value C_A3^set (Levenspiel, 1997; Fogler, 1997; Luyben, 1995; Richardson and Peacock, 1994; Smith, 1981). The variable C_AD is the disturbance concentration, and the variable C_AM is the manipulated concentration that is changed by the controller. From the system, it can be postulated that:
C_A0 = C_AM + C_AD (3)
The controller has proportional and integral action; it changes C_AM, based on the magnitude of the error (the difference between the set point concentration and C_A3) and the integral of this error. C_AM = 0.8 + K_c(E + 1/ד_f∫E(t)dt ) (4)
where:
E = CA₃^set - C_A3 (5)
K_c = Feed back Controller gain
Ί_f = Feed back Controller space- time constant (minutes). The 0.8 term is the bias value of the controller, that is, the value of C_AM at time equal zero. Numerical values of K_c = 30, ד_f = 5 minutes are used (Perry and Green, 1997; Luyben, 1995; Himmelblau, 1987; Imre, 1986). The industrial brewery effluent, has the following measured parameters; potency of hydrogen (pH), temperature (TEMP), total dissolved solid (TDS), biochemical oxygen demand (BOD), carbonate ions (CO₃^2-), calcium ions (Ca²⁺), sodium ions (Na⁺) and chloride ions (Cl ^-). Let pH =P, Temp = T₁, TDS = T₂, BOD = B, CO₃^2- = C, Ca = C₁, Na = N, Cl = C₂
To develop a model for the concentration of CO₃^2- ion as a function of the other parameters in the effluent we have:
C = f (P, T₁, T₂, B, C₁, N, C₂) = f (aP, bT₁, cT₂, dB, eC₁, f N, gC₂) (6)
where C is the dependent variable, a, b, c, d, e, f and g are the coefficients, which should be determined, while P, T₁, T₂, B C₁, N, C₂ are the independent variables of the equation. To develop the model the linear regression techniques (least square method) was applied (Stroud, 1995a,b; Carnahan et-al, 1969). Let Z represent the square of the error, between the observed value and the predicted value. Mathematically:
Z = Observed value – (aP + bT₁ + cT₂ + dB + eC₁ + fN + gC₂)² (7)
For n experimental values of P, T₁, T₂, B, C₁, N, C₂ we have:
nZ = Σ (Vi – (aP_i + bT_1i + CT_2i + dB_i + eC_1i + fN_i + gC_2i)² (8)
We could minimize the error of this regression, by finding the derivative of nZ with respect to the constant a, b, c, d, e, f, g and equating to zero. ∂(nz)/∂a = -2∑P_i(V_i–(aP_i+bT_1i+cT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0
∂(nz)/∂b = -2∑T_1i(V_i–(aP_i+bT_1i+cT_2i+dB_i+eC_2i+fN_i+gC_2i)) = 0
∂(nz)/∂c = -2∑T_2i(V_i–(aP_i+bT_2i+cT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0
∂(nz)/∂d = -2∑B_i(V_i– (aP_i+bT_1i+cT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0 (9)
∂(nz)/∂e = -2∑C_i(V_i–(aP_i+bT_2i+cT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0
∂(nz)/∂f = -2∑N_i(V_i–(aP_i+bT_1i+CT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0
∂(nz)/∂g = -2∑C2_i(V_i–(aP_i+bT_1i+CT_2i+dB_i+eC_1i+fN_i+gC_2i)) = 0
Dividing both sides by – 2 and rearranging, we obtain:
∑P_iV_i=a∑ p_i²+b∑ T_iP_i+c∑T_2iP_i+d∑B_iP_i+e∑C_iP_i+f∑N_iP_i+g∑C_2iP_i
∑T_1iV_i=a∑p_iT_i+b∑T_ii²+c∑T_2iT_1i+d∑B_iT_1i+e∑C_1iT_1i+f∑N_iT_1i+g∑C_2iT_1i
∑T₂V_i=a∑p_iT_2i+b∑T_iiT_2i+c∑T_2i²+d∑B_iT_2i+e∑C_1iT_2i+f∑N_iT_2i+g∑C_2iT_2i
∑B_iV_i=a∑P_iB_i+b∑T_1iB_i+c∑T_2iB_i+d∑B_i²+e∑C_1iB_i+f∑N_iB_i+g∑C_2iB_i (10)
∑C_1iV_i=a∑P_iC_1i+b∑T_1iC_1i+c∑T_2iC_1i+d∑B_iC_1i+e∑C_1i²+f∑N_iC_1i+g∑C_2iC_1i
∑N_iV_i=a∑P_iN_i+b∑T_1iN_i+c∑T_2iN_i+d∑B_iN_i+e∑C_1iN_i+f∑N_i²+g∑C_2iN_i
∑C_2iV_i=a∑P_iC_2i+b∑T_1iC_2i+c∑T_2iC_2i+d∑B_iC_2i+e∑C_1iC_2i+f∑N_iC_2i+g∑C_2i²

The following values were obtained from mathematical calculation (Stroud, 1995a,b; Carnahan et-al, 1969):
∑P₁² = 12276.64∙10^–8; ∑P_i T_i = 2845.59∙10^–4; ∑T_2iP_i = 930.88∙10^–4;
∑B_iP_i = 0.254900000; ∑C_1iP_i = 928.8557∙10^–4; ∑N_i P_i = 841.99000∙10^–4;
∑C_2iP_i = 174.6000∙10^–4; ∑P_iT_1i = 2845.59∙10^–4; ∑T_i²=94740.84∙10^–4;
∑T_2iT_1i = 2377.36∙10^–4; ∑B_iT_i = 0.708970000000; ∑C_iT_1i = 2641.97∙10^–4;
∑N_iT_i = 2340.830∙10^–4; ∑C_2iT_1i = 441.74∙10^–4; ∑C_1iT₂  = 845.600000∙10^–4;
∑N_iT_2i = 760.7∙10^–4; ∑C_2iT_2i =157.500000∙10^-4; ∑C_1iB_i = 0.23145000; (11)
∑N_iB_i = 0.20413000000; ∑C_2iB_i = 39.34∙10^-3; ∑N_iC_1i = 764.2002∙10^-4;
∑C_2iC_1i = 156.458∙10^-4; ∑C_2iN_i = 142.9500∙10^-4; ∑T_i² = 10179.2∙10^–8;
∑T_1i² = 94740.84∙10^-4; ∑B_i² = 762.864∙10^–6; ∑C_i² = 10124.38∙10^-8;
∑N_i² = 7144.814∙10^–8; ∑C_2i² = 351.7600∙10^-8; ∑P_iV_I = 924.2800∙10^-4;
∑T_1iV_i = 2568.1∙10^–4; ∑T_2iV_i = 839.400∙10^-4; ∑B_iV_i = 0.2297000;
∑C_1iV_i = 838.86∙10^-4; ∑N_iV_i = 763.3∙10^–4; ∑C_2iV_i = 156.33∙10^-4.
Substituting the values obtained into the equation, we have:
a(12276.64∙10^-8) + b(2845.59∙10^-4) + c(930.88∙10^-4) + d(0.2549)
+ e(928.86∙10^-4) + f(841.99∙10^-4) + g(174.60∙10^-4) = 924.28∙10^–4
a(2845.59∙10^-8) + b(94740.84∙10^-8) + c(2377.74∙10^-4) + d(0.709)
+ e(2641.97∙10^-4) + f(2340.83∙10^-4) + g(441.74∙10^-4) = 2568.1∙10^–4
a(930.88∙10^-4) + b(2377.36∙10^-4) + c(10179.2∙10^-8) + d(0.2326)
+ e(845.6∙10^-4) + f(760∙10^-4) + g (157.5∙10^-4) = 839.4∙10^–4 (12)
a(0.2549) + b(0.709) + c(0.2326) + d(762.864∙10^-6)
+ e (0.23145) + f(0.20413) + g (39.34 10^-3) = 0.2297
a(928.86∙10^-4) + b(2621.97∙10^-4) + c(845.6∙10^-4) + d(0.23145)
+ e(10124.38∙10^-8) + f(764.202∙10^–4) + g(156.458∙10^{- 4}) = 838.86∙10^–4
a(841.99∙10^-4) + b(2340.83∙10^-4) + c(760.7∙10^-4) + d(0.20413)
+ e(764.202∙10^-4) + f(7144.814∙10^-8 ) + g(142.95∙10^-4) = 763.3∙10^–4
a(174.60∙10^-4) + b(441.74∙10^-4) + c(157.5∙10^-4) + d(39.34∙10^-3)
+ e(156.458∙10^-4) + f(142.95∙10^-4) + g (351.76∙10^-8) = 156.33∙10^–4

A computer program, coded in C++ was developed to solve the 7´7 matrices and the resulting model is:
C = 0.163488P + 0.0514255T₁ + 0.166722T₂ + 0.0596487B
+ 0.182077C₁ + 0.87642N + 0.821217C₂ (13)


Results and Discussion

The effluent from the brewery was discharged at different points such as the tank cellars, bottling hall, brew house and filter-room. The effluent analyzed in this study was however obtained from the combined effluent discharge collection point. The analytical results are presented in Tables 1, 2 and 3.
Table 1. Analysis of effluent discharge from brewery industry (1999)

	pH	TEMP (°C)	TDS (kg/dm3)	BOD (kg/dm3)	CO32- (kg/dm3)	Ca2+ (kg/dm3)	Na+ (kg/dm3)	Cl- (kg/dm3)
Jan	9.60	27.60	8.211	2.35∙10-3	13.8421	8.480	8.106	1.477
Feb	9.00	28.20	8.120	2.30∙10-3	13.6687	8.146	8.125	1.523
Mar	9.50	26.50	8.000	2.35∙10-3	14.9868	8.742	8.636	1.612
Apr	9.20	26.00	9.512	2.40∙10-3	14.8273	8.102	8.064	1.585
May	9.30	26.00	8.167	2.30∙10-3	13.9182	8.261	8.093	1.532
Jun	9.40	24.50	8.000	2.00∙10-3	13.7231	7.992	7.602	1.632
Jul	9.40	26.00	8.261	2.30∙10-3	13.5698	8.266	8.611	1.723
Aug	9.10	26.50	8.082	2.35∙10-3	14.8955	8.361	8.626	1.439
Sep	9.00	24.00	8.664	2.44∙10-3	14.0688	8.073	8.0423	1.321
Oct	9.00	24.00	8.164	2.40∙10-3	13.5890	8.042	7.110	1.732
Nov	9.20	26.00	9.026	2.40∙10-3	13.9936	8.364	8.210	1.521
Dec	9.10	24.50	8.164	2.33∙10-3	13.9814	8.772	8.116	1.621

The effluent was analyzed based on several parameters of pH, temperature and ionic content (CO₃^2-, TDS, Ca, Na, Cl, BOD). From the empirical results presented on Tables 1 to 3, it was observed that the determinant ion is CO₃^2-, because of its high concentration and its high acidity.
Table 2. Analysis of effluent discharge from brewery industry (2000)

	pH	TEMP (°C)	TDS (kg/dm3)	BOD (kg/dm3)	CO32- (kg/dm3)	Ca2+ (kg/dm3)	Na+ (kg/dm3)	Cl- (kg/dm3)
Jan	9.40	26.50	8.575	2.37∙10-3	13.0842	8.183	8.105	1.532
Feb	9.10	26.00	8.324	2.33∙10-3	13.9271	8.176	8.173	1.561
Mar	9.70	27.00	8.175	2.40∙10-3	13.9866	8.233	8.621	1.663
Apr	9.30	24.00	8.025	2.30∙10-3	13.1182	8.136	7.189	1.419
May	9.10	25.00	8.000	2.36∙10-3	13.3970	8.491	7.326	1.567
Jun	9.10	28.00	8.000	2.37∙10-3	13.774	8.247	7.961	1.668
Jul	9.20	26.30	8.232	2.40∙10-3	13.0736	8.566	6.119	1.589
Aug	9.40	24.50	8.727	2.20∙10-3	13.0180	8.373	6.327	1.772
Sep	9.00	23.00	8.632	2.25∙10-3	13.0000	8.392	6.331	1.699
Oct	9.30	26.00	8.442	2.40∙10-3	13.2941	8.626	6.861	1.427
Nov	9.10	28.00	8.665	2.35∙10-3	13.5084	8.286	7.544	1.489
Dec	9.10	24.00	8.361	2.33∙10-3	13.5126	8.725	7.612	1.551

The data presented are combined effluent discharge data for three different years (1999, 2000, and 2001), in all cases the carbonate ion was observed to dominate in concentration. Therefore part of the aims of this work is to develop appropriate techniques to conveniently reduce the concentration of this ion. The process chosen for this task was characterized by two neutralization reactions occurring simultaneously in the reaction vessel as given by Equation 1 and 2. (Meyer, 1992a,b,c; Luyben, 1990, Austin, 1984). The principal reactions of consideration are Equation 1, where the model equation is represented as:
A + B → C (14)
where A is Calcium ion; B is Carbonate ion; C is Calcium carbonate. The limiting reactant is B and its conversion is of interest in this research work. Appropriate software coded in Visual Basic was developed to monitor the concentration of the carbonate ions in the system, the results are presented on Tables 7, 8 and 9. From the Tables it was observed that there was a change in concentration (conversion) across the tanks in series, this change was attributed to the reaction given by Equation 1. Table 3. Analysis of effluent discharge from brewery industry (2001)

	pH	TEMP (°C)	TDS (kg/dm3)	BOD (kg/dm3)	CO32- (kg/dm3)	Ca2+ (kg/dm3)	Na+ (kg/dm3)	Cl- (kg/dm3)
Jan	9.70	28.00	8.164	2.30∙10-3	13.973	8.274	7.994	1.6114
Feb	8.90	28.30	8.180	2.30∙10-3	13.083	8.354	8.074	1.6114
Mar	9.60	26.00	7.990	2.40∙10-3	14.886	8.399	8.883	1.7390
Apr	9.10	24.00	9.410	2.44∙10-3	12.818	7.832	6.495	1.5650
May	9.30	26.40	8.082	2.36∙10-3	12.545	8.286	6.514	1.4450
Jun	9.40	24.00	7.999	1.71∙10-3	14.922	8.626	9.061	1.7390
Jul	9.70	27.90	7.999	2.30∙10-3	14.367	8.399	8.291	1.5890
Aug	9.00	26.40	8.263	2.35∙10-3	13.262	8.399	8.291	1.6114
Sep	8.80	22.50	9.163	2.33∙10-3	12.471	8.853	6.119	1.5470
Oct	9.10	24.90	8.090	2.41∙10-3	12.584	8.286	6.514	1.5730
Nov	9.30	26.30	9.412	2.37∙10-3	12.604	8.286	6.317	1.4320
Dec	8.90	23.10	8.131	2.35∙10-3	13897	8.626	8.488	1.2920

For the first case the feed enters the first reactor with the following concentrations: C_A2 at 10 kg/dm³, C_A3 at 0.1 kg/dm³ while C_AM is at 0.8 kg/dm³ (which is the bias value of the controller at time zero, T = 0), the feedback controller used for this process has proportional and integral action (PI controller), it changes C_AM based on the magnitude of the error. The value of the set point, C_A3^set was at 0.1 kg/dm³, the controller looks at the product concentration leaving the third Tank, C_A3, and makes adjustments in the inlet concentration to the first reactor, C_A0, in order to keep C_A3 near its set point C_A3^set. The variable C_AD is the disturbance concentration and the variable C_AM is the manipulated concentration that was changed by the controller (Equation 3). For this particular simulation (applying Equations 3 and 4), the following values were used; K_c at 30, ד_f at 5minutes. Critical observation shows that the output concentration of the system (C_A3) approaches the set point concentration (C_A3^set), as shown Table 7 to 9. The difference between the set point value and C_A3 in each case is small, this was attributed to the reactor design specification and reactor operating condition (Luyben, 1990; Himmelblau, 1987). Generally, the whole essence of the process set-up is to neutralize the concentration of the carbonate ions to values low enough, so as not to be toxic to the environment, and to a reasonable extent this was achieved. Tables 1-3 gives the analysis of the effluent discharge from the brewery industry for three consecutive years. The pH values for the years are slightly high, with a minimum of 9.00 for the first two years and 8.90 for the third year. On the general the pH were mostly alkaline from the discharge. The temperature was within the set limits by the Federal Environmental Protection Agency (FEPA). The limits of the other parameters were also within the acceptable limits. The facts is that all these compounds present in the effluent could be hydrolyzed by water, that is, the ions of these compounds could be exchanged with water molecules (Odigure and Adeniyi, 2002; Karapetyant and Drakin, 1981). The carbonate ions concentrations from Table 1-3 are relatively higher than the other parameters. Comparative values for the concentration of carbonate between the empirical and simulation are presented in Tables 4 to 6. The highest deviation for 1999 was 8.70%, for 2000 was 5.50% and 1.70% for 2001. The deviation of the simulated values from that of the empirical could be attributed to certain limitations placed during model development. Table 4. Comparative values for empirical and simulation of CO₃^2- (1999)

	Empirical (kg/dm3)	Simulated (kg/dm3)	Deviations	%Errors
Jan	13.8421	14.2193	-0.3772	2.73
Feb	13.6687	14.1303	-0.4616	3.38
Mar	14.9868	14.6850	0.4129	2.76
Apr	14.8273	14.4144	0.4129	2.78
May	13.9182	14.0744	-0.1562	1.12
Jun	13.7231	13.5885	0.1346	0.98
Jul	13.5698	14.7508	-1.181	8.70
Aug	14.8955	14.4622	0.4333	2.91
Sep	14.0688	13.7540	0.3148	2.24
Oct	13.5890	13.1848	0.4042	2.97
Nov	13.9936	14.3135	-0.3199	2.29
Dec	13.9814	14.1535	-0.1689	1.21

That is, apart from these seven parameters considered there could be other factors, which contributed to the level of carbonates in the effluent. One is the interactions, which the effluents would have undergone during the process of flowing through the discharge path.
Table 5. Comparative values for empirical and simulation of CO₃^2- (2000)

	Empirical (kg/dm3)	Simulated (kg/dm3)	Deviations	%Errors
Jan	13.0842	12.7240	0.3602	2.75
Feb	13.9271	14.0867	-0.1596	1.15
Mar	13.9866	14.7578	-0.7712	5.50
Apr	13.1182	13.0387	0.0795	0.60
May	13.3970	13.3595	0.0375	0.30
Jun	13.7740	14.1089	-0.3349	2.40
Jul	13.0736	12.4553	0.6183	4.70
Aug	13.0180	12.7754	0.2426	1.90
Sep	13.0000	12.5641	0.4359	3.40
Oct	13.2941	13.0194	0.2747	2.10
Nov	13.5084	13.7144	-0.2060	1.50
Dec	13.5126	13.6484	-0.1358	1.00

Table 6. Comparative values for empirical and simulation of CO₃^2- (2001)

	Empirical (kg/dm3)	Simulated (kg/dm3)	Deviations	%Errors
Jan	13.973	14.2132	-0.2373	1.70
Feb	13.083	12.9527	0.1303	1.00
Mar	14.886	14.9830	-0.097	0.65
Apr	12.818	12.6945	0.1235	0.96
May	12.545	12.6300	-0.085	0.68
Jun	14.922	15.0447	-0.1227	0.82
Jul	14.367	14.4452	-0.0782	0.54
Aug	13.262	13.3709	-0.1089	0.82
Sep	12.471	12.3688	0.1022	0.82
Oct	12.584	12.6266	-0.0426	0.34
Nov	12.604	12.6633	-0.0593	0.47
Dec	13.897	14.0094	-0.1124	0.81

Consequently, pollutants present in water could seriously affect the resultant carbonate concentration. The extent of acidification or alkalization of the solution by pollutants is dependent, on not only the chemical nature of the compounds present, but also the prevailing technological conditions (Odigure and Adeniyi, 2002). From the calculated coefficients, it could be seen that the carbonate concentration is most affected by the sodium ions and least by the temperature. Tables 7- 9 gives the time-concentration value from simulation of the months of January, March and June of the year 2001.
Table 7: Time – Concentration data from simulation (Jan. 2001)

Run	Time (min)	CA1 (kg/dm3)	CA2 (kg/dm3)	CA3 (kg/dm3)	CAM (kg/dm3)
1	0.00	13.0830	10.0000	0.1000	0.8000
2	0.50	1.7949	7.5767	1.8086	53.3328
8	3.51	9.2574	1.9208	1.6843	48.9541
12	5.51	1.0135	2.9864	1.1401	32.2791
19	9.01	7.1980	1.2045	0.3007	13.1300
20	9.51	5.6166	2.0257	0.1556	3.2070

From Table 7 at time zero, the conversion of CO₃^2- was 0.8 kg/dm³ (which is the bias value of the controller), at T=0.5 minutes, the conversion was 53.33278 kg/dm³, showing that with increase in time the manipulated variable concentration decreases proportionately toward the set point. The same pattern was observable in Table 8 and 9. Table 8: Time – Concentration data from simulation (Mar. 2001)

Run	Time (min.)	CA1 (kg/dm3)	CA2 (kg/dm3)	CA3 (kg/dm3)	CAM (kg/dm3)
1	0.00	14.8860	10.0000	0.1000	0.8000
2	0.50	2.8012	7.8471	1.8242	54.3856
8	3.51	9.1626	2.1493	1.7621	50.7102
12	5.51	0.7137	3.1506	1.1611	32.9314
19	9.01	7.4191	1.1768	0.3364	14.0958
20	9.51	5.9060	2.0619	0.1355	0.5531

Table 9: Time – Concentration data from simulation (Jun. 2001)

Run	Time (min.)	CA1 (kg/dm3)	CA2 (kg/dm3)	CA3 (kg/dm3)	CAM (kg/dm3)
1	0.00	14.9220	10.0000	0.1000	0.8000
2	0.50	2.8213	7.8525	1.8431	54.4067
8	3.51	9.1607	2.1539	1.7638	50.7430
12	5.51	0.7078	3.1538	1.1655	32.9445
19	9.01	7.4236	1.1762	0.3371	14.1151
20	9.51	5.9118	2.0626	0.1351	0.5648

Table 10 gives the percentage error deviation analysis of the simulated concentration for the third tank to that of the set point. This gives the deviation error analysis for the process control model. The highest deviation was 38% at the fifth run.
Table 10: Error deviation analysis for process control model

Run	CA3	CA3set	Error Deviation	% Error Deviation
1	0.14565	0.1	0.04565	31
2	0.15564	0.1	0.05560	36
3	0.13548	0.1	0.03550	26
4	0.15852	0.1	0.05852	37
5	0.16156	0.1	0.06156	38
6	0.13508	0.1	0.03510	26
7	0.14126	0.1	0.04130	29
8	0.15357	0.1	0.05360	35

Conclusions

From the empirical model developed, the constituent parameter with greatest influence was Na⁺ with a coefficient of 0.87642, while that with the least influence, was the temperature with a coefficient of 0.0514255. This study, in addition, seeks to reduce the concentration of carbonate ion to an acceptable degree, so as not to degrade the environment. To some extent, this could be said to be have been achieved, since the concentration was lowered from 13.973 to 0.145648108458162 in the first case. Under ideal operating conditions, the output concentration from the system should converge at the set point (i.e. C_A3=C_A3^set). In such a case, no error is generated, but because of differences in the design specifications and process parameters, such errors are inevitable as shown in the results presented. Thus the proposed model could be used to predict the concentration of carbonate ions from brewery effluent with similar operating conditions.

Acknowledgment

The help rendered by Mr. C. M. Dikwal of the Department of Chemical Engineering, Federal University of Technology, Minna, Nigeria is highly appreciated.

References

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Binomial Distribution Sample Confidence Intervals Estimation 1. Sampling and Medical Key Parameters Calculation

Tudor DRUGAN^a, Sorana BOLBOACĂ^a*, Lorentz JÄNTSCHI^b, Andrei ACHIMAŞ CADARIU^a
a “Iuliu Haţieganu” University of Medicine and Pharmacy, Cluj-Napoca, Romania
^b Technical University of Cluj-Napoca, Romania
* corresponding author, sbolboaca@umfcluj.ro


Abstract
The aim of the paper was to present the usefulness of the binomial distribution in studying of the contingency tables and the problems of approximation to normality of binomial distribution (the limits, advantages, and disadvantages). The classification of the medical keys parameters reported in medical literature and expressing them using the contingency table units based on their mathematical expressions restrict the discussion of the confidence intervals from 34 parameters to 9 mathematical expressions. The problem of obtaining different information starting with the computed confidence interval for a specified method, information like confidence intervals boundaries, percentages of the experimental errors, the standard deviation of the experimental errors and the deviation relative to significance level was solves through implementation in PHP programming language of original algorithms. The cases of expression, which contain two binomial variables, were separately treated. An original method of computing the confidence interval for the case of two-variable expression was proposed and implemented. The graphical representation of the expression of two binomial variables for which the variation domain of one of the variable depend on the other variable was a real problem because the most of the software used interpolation in graphical representation and the surface maps were quadratic instead of triangular. Based on an original algorithm, a module was implements in PHP in order to represent graphically the triangular surface plots. All the implementation described above was uses in computing the confidence intervals and estimating their performance for binomial distributions sample sizes and variable.
Keywords
Confidence Interval; Binomial Distribution; Contingency Table; Medical Key Parameters; PHP Applications


Introduction
The main aim of a medical research is to generate new knowledge which to be used in healthcare. Most of the medical studies generate some key parameters of interest in order to measure the effect size of a healthcare intervention. The effect size can be measure in a variety of way such as sensibility, specificity, overall accuracy and so on. Whatever the measure used are, some assessment must be made of the trustworthiness or robustness of the finding [1]. The finding of a medical study can provide a point estimation of effect and nowadays this point estimation had a confidence interval that allows us interpreting correctly the point estimation of an interest parameter. The formal definition of a confidence interval is a confidence interval gives an estimated range of values that is likely to include an unknown population parameter, the estimated range being calculates from a given set of sample data. If independent sample are take repeatedly from same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. Confidence intervals are usually computes so that this percentage is 95%. However, we can produce 90%, 99%, 99.9% confidence intervals for an unknown parameter [2]. In medicine, we work with two types of variables: qualitative and quantitative, variables that can be classifies into theoretical distributions. Continuous variables follow a normal probability distribution, Laplace-Gauss distribution and discrete random variables follow a Binomial distribution [3]. The aim of this series of article was to review the confidence intervals used for different medical key parameters and introduce some new methods in confidence intervals estimation. The aim of the first article was to present the basic concept of confidence intervals and the problem, which can occur in confidence intervals estimation.

Binomial Distribution and its Approximation to Normality
The normal distribution was first introduced by De Moivre in an unpublished memorandum later published as part of [4] in the context of approximating certain binomial distribution for large n. His results were extends by Laplace and are now called the Theorem of De Moivre-Laplace. Confidence interval estimations for proportions using normal approximation have been commonly uses for analysis of simulation for a simple fact: the normal approximation was easier to use in practice comparing with other approximate estimators [5]. We decided to work in our study with binomial distribution and its approximation to normality. Let’s saw how binomial distribution and its approximation to normality works! If we had from a sample size (n) a variable (X, 0 ≤ X ≤ n) which follows a binomial distribution the probability of obtaining the Y value (0 ≤ Y ≤ n) from same sample is:
(1)
The mean and the variance for the binomial variable are:
, (2)
The apparition probability of a normal variable Y (0 ≤ Y ≤ n) which has a mean M(n,X) and a standard deviation Var(n,X) is:
(3)
The approximation to normality for a binomial variable used the values of the mean and of the dispersion. Replacing the mean and the dispersion in binomial distribution formula, we obtain the next formula for 0 < X < n:
(4)
The approximation error of binomial distribution of a variable Y (0 ≤ Y ≤ n) with to normal distribution is:
(5)
Because the probability of Y decreasing with straggle of the expected value X, for mark out this digression we can adjust the expression above to zero for P_B(n,X,Y) < 1/n (in this case the differences can not be remarked for a extraction from a binomial sample):
(6)
In figure 1 was represent the approximation error of a binomial distribution with a normal distribution (using the formula above) for X/n = 1/2 where n = 10, 30, 100 and 300 (Err_c(n,n/2,Y)).
 
 
Figure 1. The approximation error of the binomial distribution with a normal distribution for X/n=1/2 where n=10, 30, 100 and 300.
If we chouse to represent the errors of the approximation of a binomial distribution with a normal distribution for X/n=1/10 where n=10, 30, 100 and 300 (Err_c(n,n/10,Y)) we obtain the below graphical representation (figure 2).


Figure 2. The approximation error of the binomial distribution with a normal distribution for X/n=1/10 where n=10, 30, 100 and 300.
The experimental errors for the 5/10 fraction (X=5, n=10) with normal approximation induce an underestimation for X = 3 and X = 7 with a sum of 7.8‰ and an overestimation for X = 4, X = 5 and X = 6 with a percent sum of 8.3‰ which compared with the choused significance level α = 5% represent less than 10%. However, the error variation shows us that the normal approximation induce a decreasing of the confidence interval for a proportion being considered more frequently the values neighbor to the 5 than the extremities values. Graphical representation for the variation of the errors sum of positive estimation (for the values away to the measured values) and the variation of the error sum of negative estimation (for the values closed to the measured values) in logarithmical scale (n, and the errors sum) was presented in figure 7.

Figure 3. Logarithmical variation of the sum of positive (log(sp)) and negative(log(sm)) estimation errors depending on log(n) function for X/n = 1/2 fraction
Looking at the graphical representation from the figure 7 it can be observe that the variation of the errors sum was almost linear and inverse proportional with n. For the proportion X/n = 1/2 and generally for medium proportion the normal approximation is good and its quality increase with increasing of sample size (n). Moreover, to the surrounding of 1/2 the symmetry hypothesis of the confidence estimation induce by the normality approximation work perfectly. Looking at the variation of the positive errors induced by the normal approximation for the proportion in form of X/n = 1/10 depending on n (see figure 4), it can be remarked that the abscissa of maximum point of deviation relative to binomial distribution are found at lower n in the vicinity of the extreme values. Thus, for n = 10, and X = 1 the error reach the maximum value of 0.108 at 0. The abscissa is moving towards X value with increasing of n (for n = 300, and X = 30 the error reach the maximum value of 0.0283 at 25). For the variation of the negative errors induces by the normal approximation of the proportion of type X/n=1/10 depending on n, we can remark that the abscissa of maximum point of deviation depending on binomial distribution are found at lower n in the vicinity of the extreme values. Thus, for n = 10, and X = 3 the error reach the maximum value of 0.05 at 0. The abscissa is moving towards X value with increasing of n (for n = 300, and X = 30 the error reach the maximum value of 0.0260 at 34). More over, the approximation errors depending on X value is asymmetrical, and there was not any tendency for symmetry with increasing of n.

Figure 4. Logarithmic variation of the sum of values less than X (log(si)) and greater than X (log(ss)) depending on log(n) for X/n = 1/10 fraction
Because the sum of errors for values smaller than X were always greater than the sum of errors for the values greater than X, the confidence interval obtained base don normality approximation it will be displaced to the right (there were generate more approximation errors on the left side).


Figure 5. The error of binomial distribution approximate with a normal distribution with X/n = 1/n fraction express through Err_c(n,1,Y) function at n = 10, n = 30, n = 100 and n = 300
The error had a maximum value of 10.8% for n = 10 and X = 1 into point 0, value which was greater twice than the accepted significance level α = 5% (!). A maximum value of 2.6% were obtained twice consecutively for n = 30, X = 3 into the 1 and 2 points and a maximum value of 2.5% into the point 4, that was half reported to the significance level α = 5% (!); which represent a sever deviation. For large n the sum of the approximation error there was a tendency to linearity inverse proportional with n.

Figure 6. The variation of the error estimation sum for X/n = 1/n fraction
The graphical representations from the figure 5 and 6 reveal that for small proportions (X/n = 1/n) the sum of approximation errors were increasing with n. The sum of the approximation errors always exceeded the significance level (α = 5% (!)) for n > 50, which means that the real error threshold using normality hypothesis will exceeded 10%, the error being obtained in the favor of the greater values for the X.

Implementation of the Mathematical Calculation in MathCad
The experimental described in the previous section was possible after a MathCad implementation. The results obtained running the MathCad program were imports in Microsoft Excel were the graphical representations were creates. In MathCad were implements the next functions:
· n combination taking to k: was necessary a logarithmical expression because we were working with large sample size and for n = 300 we could not used the classical formula C(n,k) = n!/((n-k)!k!)
(7)
(8)
(9)
The calculation of Y/n apparition into a binomial distribution by n sample size and X/n proportion:
(10)
The computing of the apparition of Y/n into a normal distribution sample by volume n, mean X/n and variance X(n-X)/n:
(11)
The differences between the binomial and normal distribution:
(12)
Initializations (needed for defining the series of values):
 (values which were consecutively changed with 10,1; 10,5; 30,1; 30,3; 30,15; 100,1; 100,10; 100,50; 300,1; 300,30; 300,150)

The computing of the series of corrected differences (graphically represented in figure 1, 2 and 5):
(13)
The construction of the series of the positive sum of probabilities differences until to X (graphically represented in figure 3)
(14)
The construction of the series of the positive sum of probabilities differences for the values less than X/n (graphically represented in figure 4)
(15)
 - display the values of the sum for each pair of data
The calculation of the differences probabilities for every parameters:
 display the values of the sum for each pair of data (16)
Was also used a File Read/Write component (linked with dBNC_Y) in order to export the results into *.txt file from where the data were imported in Microsoft Excel where the graphical representation were drawing.

Generating a Binomial Sample using a PHP Program
In order to evaluate the confidence intervals for medical key parameters we decide to work with binomial samples and variables. In PHP was creates a module with functions that generates binomial distribution probabilities. Each by another, was implements functions for computing the binomial coefficient of n vs. k (bino0 function), the binomial probability for n, X, Y (bino1, bino2, and bino3 functions), drawing out a binomial sample of size n_es (bino4 function). Finally, array rearrangement of binomial sample values staring with position 0 and saving in the last element the value of rearrangement postponement are makes by bino5 function. The used functions are:
function bino0($n,$k){ $rz = 1;
 if($k<$n-$k) $k=$n-$k;
 for ($j=1;$j<=$n-$k;$j++)
$rz *= ($k+$j)/$j;
 return $rz; }
function bino1($n,$y,$x){ return pow($x/$n,$y)*pow(1-$x/$n,$n-$y)*bino0($n,$y); }
function bino2($n,$y,$x){
 if(!$x) if($y) return 0; else return 1;
 $nx=$n-$x; $ny=$n-$y;
 if(!$nx) if($ny) return 0; else return 1;
 $ret=0;
 if($y){
$ret += log($x)*$y;
if($ny){
$ret += log($nx)*$ny;
if($ny>$y) $y=$ny;
for($k=$y+1;$k<=$n;$ret+=log($k++));
for($k=2;$k<=$n-$y;$ret-=log($k++)); }
 } else $ret += log($nx)*$ny;
 return exp($ret-log($n)*$n); }
function bino3($n,$y,$x){
if(!$x) if($y) return 0; else return 1;
 $x/=$n; $nx=1-$x; $ny=$n-$y;
 if(!$nx) if($ny) return 0; else return 1;
 $ret=0;
 if($y){
$ret += log($x)*$y;
if($ny){
$ret += log($nx)*$ny;
if($ny>$y) $y=$ny;
for($k=$y+1;$k<=$n;$ret+=log($k++));
for($k=2;$k<=$n-$y;$ret-=log($k++)); }
 } else $ret += log($nx)*$ny;
 return exp($ret); }
function bino4($n,$k,$n_es){
 for($j=0;$j<=$n;$j++){
if($j>15) $c[$j]=bino1($n,$j,$k); else $c[$j]=bino3($n,$j,$k);
$c[$j]=round($c[$j]*$n_es);
if(!$c[$j])unset($c[$j]); }
 return $c; }
function bino5($N,$k,$f_N){
 $d=bino4($N,$k,$f_N);
 $j=0;
 foreach($d as $v) $r[$j++]=$v;
 foreach($d as $i => $v) break;
 $r[count($r)]=$i;
 return $r; }
Computing the confidence intervals for a binomial variable is time consuming. In this perspective, an essential problem was optimizing the computing time. We studied comparative the time needed for generating the binomial probabilities using the bino1 function, the bino2 function, and the bino3 function. We developed a module which to collect and display the execution time. In the appreciation of the execution time, we used the internal clock of the computer. The module that collects and displays the execution time is:
$af="N\tBino1\tBino2\tBino3\r\n";
for($n=$st;$n<$st+1;$n++){
 $t1=microtime(TRUE);
 for($x=0;$x<=$n;$x++) for($y=0;$y<=$n;$y++)
$b=bino1($n,$y,$x);
 $t2=microtime(TRUE);
 for($x=0;$x<=$n;$x++) for($y=0;$y<=$n;$y++)
$b=bino2($n,$y,$x);
 $t3=microtime(TRUE);
 for($x=0;$x<=$n;$x++) for($y=0;$y<=$n;$y++)
$b=bino3($n,$y,$x);
 $t4=microtime(TRUE);
 $af.=$n."\t";
 $af.=sprintf("%5.3f",log((10000*($t2-$t1))))."\t";
 $af.=sprintf("%5.3f",log((10000*($t3-$t2))))."\t";
 $af.=sprintf("%5.3f",log((10000*($t4-$t3))))."\r\n";}
echo($af);
Table 1 contains the logarithmical values of the executions time for n: 3, 10, 30, 50, 100, 150, and 300 obtained by the bino1, bino2, and bino3 functions.

n	Bino1	Bino2	Bino3
3	1.92	1.573	1.406
10	3.975	4.021	3.956
30	6.441	6.763	6.74
50	7.731	8.149	8.137
100	9.576	10.108	10.101
150	10.706	11.28	11.278
300	12.691	13.314	13.314

Table 1. The execution time (seconds) express in logarithmical scale needed for generating binomial probabilities with implemented function
The differences between executions time express in logarithmical scale for the functions bino1and bino3 and respectively for the functions bino2 and bino3 were uses in order to obtain the graphical representations from the figure 7. Looking at the execution time ratio for the bino1 and bino3 function (the Bino2-Bino3 curve, figure 7) we can remark that optimizing parameter with X/n instead of X and (n-X)/n instead of (n-X) bring an increasing performance; this increased performance was cancel with increasing of n (the logarithmical difference was almost 0 for n >50). Looking at the execution time ratio for the bino2 and bino3 functions (the Bino2-Bino3 curve, figure ) we can remark that with increasing of n the bino3 function obtained the best performance comparing with bino1 and bino2 (the differences between execution time express in logarithmical scale became less than 0 beginning with n ≈ 15). Based on above observations, was implements for the binomial probabilities estimation the bino4 function.

Figure 7. Binomial probabilities execution time for bino1, bino2, and bino3 functions

Medical Key Parameters on 2X2 Contingency Table
For the 2X2 contingency table data and based on the type of the medical studies we can compute some key parameters used for interpreting the studies results. Let us suppose that we have a contingency table as the one in the table 2.

		Result/Disease (Characteristic B)
		Positive	Negative	Total
Treatment / Risk factor / Diagnostic Test (Characteristic A)	Positive	a	b	a+b
	Negative	c	d	c+d
	Total	a+c	b+d	a+b+c+d

Table 2. Hypothetical 2X2 contingency table
In table 2, a denotes the real positive cases (those who have both characteristics), b the false positive cases (those who have just the characteristic A), c the false negative cases (those who have the characteristic B) and d the real negative cases (those who have neither characteristic). We classified the medical key parameters based on the mathematical expressions (see table 3). This classification was very useful in confidence intervals assessment for each function, and for implementing the function in PHP. The PHP functions for the mathematical expression described in the table 3 had the next expressions:
function ci1($X,$n){
 return $X/$n; }
function ci2($X,$n){
 if($X==$n) return (float)"INF";
 return $X/($n-$X); }
function ci3($X,$n){
 if(2*$X==$n) return (float)"INF";
 return $X/(2*$X-$n); }
function ci4($X,$m,$Y,$n){
 if(($X==$m)||($Y==0)) return (float)"INF";
 return $X*($n-$Y)/$Y/($m-$X); }
function ci5($X,$m,$Y,$n){
 return $Y/$n-$X/$m; }
function ci6($X,$m,$Y,$n){
 return abs(ci5($X,$m,$Y,$n)); }
function ci7($X,$m,$Y,$n){
 $val=ci6($X,$m,$Y,$n);
 if(!$val) return (float)"INF";
 return 1/$val; }
function ci8($X,$m,$Y,$n){
 if($Y==0) return (float)"INF";
 return ci1($X,$m)/ci1($Y,$n); }
function ci9($X,$m,$Y,$n){
 if($Y==0) return (float)"INF";
 return abs(1-ci1($X,$m)/ci1($Y,$n)); }

No	Mathematical expression and PHP function	Name of the key parameter	Type of study
1	, ci1	Individual Risk on Exposure Group and NonExposure Group	Risk factor or prognostic factor studies [6], [7]
		Experimental and Control Event Rates Absolute Risk in Treatment and Control Group	Therapy studies [8], [9]
		Sensitivity; Specificity; Positive and Negative Predictive Value; False Positive and Negative Rate; Probability of Disease, Positive and Negative Test Wrong; Overall Accuracy; Probability of a Negative and Positive Test	Diagnostic studies [10], [11], [12]
2	, ci2	Post and Pre Test Odds	Diagnostic studies 7, 9, 10]
3	, ci3	Post Test Probability	Diagnostic studies [9, 10]
4	, ci4	Odds Ratio	Risk Factor studies [6, 7]
5	, ci5	Excess risk	Risk Factor studies [5]
6	, ci6	Absolute Risk Reduction and Increase, Benefit Increase	Therapy studies 7, 9]
7	, ci7	Number Needed to Treat and Harm	Therapy studies [7, 9]
8	, ci8	Relative Risk	Studies of Risk Factor [5, 6]
		Positive and Negative Likelihood Ratio	Diagnostic studies [7, 9, 10]
9	, ci9	Relative Risk Reduction, Increase and Benefit Increase	Therapy studies [7, 9]

Table 3. Mathematical expressions and PHP functions for medical key parameters

Estimating Confidence Intervals using a PHP program
In order to be able to check out if that the experimental numbers belong to confidence interval were implements in PHP two functions, one named out_ci_X for one-variable functions (depending on X) and the other, named out_ci_XY for two-variables functions (depending on X, Y). The functions had the expressions:
function out_ci_X(&$cif,$XX,$X,$n,$z,$a,$f,$g){
 $ci_v=ci($cif,$g,$X,$n,$z,$a);
 if(!is_finite($ci_v[1])) return 0;
 if(!is_finite($ci_v[0])) return 0;
 $ci_v=ci($cif,$g,$XX,$n,$z,$a);
 if(!is_finite($ci_v[1])) return 0;
 if(!is_finite($ci_v[0])) return 0;
 $fx=$f($X,$n);
 if(!is_finite($fx)) return 0;
 return (($fx<$ci_v[0])||($fx>$ci_v[1]));}
function out_ci_XY(&$cif,$XX,$X,$YY,$Y,$m,$n,$z,$a,$f){
 $ci_v=$cif($XX,$m,$YY,$n,$z,$a);
 if(!is_finite($ci_v[0])) return 0;
 if(!is_finite($ci_v[1])) return 0;
 $fxy=$f($X,$m,$Y,$n);
 if(!is_finite($fxy)) return 0;
 return (($fxy<$ci_v[0])||($fxy>$ci_v[1]));}
The function est_ci_er compute the confidence interval boundaries, the experimental errors, the mean, and the standard deviation for the expression with one variable (X); the est_ci2_er was the function for the two-variable expressions (X, Y):
function est_ci_er($z,$a,$c_i,$g,$f,$ce){
 if($ce=="ci") est_X(z,a,$c_i,$f,$g);
 if($ce=="er") est_N(z,a,$c_i,$f,$g,1);
 if($ce=="me") est_N(z,a,$c_i,$f,$g,2);
 if($ce=="va") est_N(z,a,$c_i,$f,$g,3);
 if($ce=="mv") est_N(z,a,$c_i,$f,$g,6);}
function est_ci2_er($z,$a,$c_i,$f,$ce){
 if($ce=="ci") est_X2(z,a,$c_i,$f);
 if($ce=="er") est_N2(z,a,$c_i,$f);
 if($ce=="me") est_MN(z,a,$c_i,$f,2);
 if($ce=="va") est_MN(z,a,$c_i,$f,3);
 if($ce=="mv") est_MN(z,a,$c_i,$f,6);}
The est_X compute all the confidence intervals for the one-variable functions (depending on X) while est_X2 compute the confidence intervals for two-variable functions (depending on X, Y):
function est_X($z,$a,$c_i,$f,$g){
 echo("N\tX\tf(X)");
 for($i=0;$i<count($c_i);$i++){
echo("\t".$c_i[$i]."L");
echo("\t".$c_i[$i]."U"); }
 for($n=N_min;$n<N_max;$n++){
for($X=0;$X<=$n;$X++){
for($i=0;$i<count($c_i);$i++)
$ci[$i] = ci($c_i[$i],$g,$X,$n,$z,$a);
$fX = $f($X,$n);
echo("\r\n".$n."\t".$X."\t".trim(sprintf("%6.3f",$fX)));
for($k=0;$k<count($c_i);$k++){
echo("\t".trim(sprintf("%6.3f",$ci[$k][0])));
echo("\t".trim(sprintf("%6.3f",$ci[$k][1]))); } }}}
function est_X2($z,$a,$c_i,$f){
 echo("X\tY\tf(X,Y)");
 for($i=0;$i<count($c_i);$i++){
echo("\t".$c_i[$i]."L");
echo("\t".$c_i[$i]."U"); }
 for($n=N_min;$n<N_max;$n++){
$m = $n;
for($X=0;$X<=$m;$X++){
for($Y=0;$Y<=$n;$Y++){
for($i=0;$i<count($c_i);$i++){
$g = $c_i[$i];
$ci[$i] = $g($X,$m,$Y,$n,$z,$a); }
$fXY = $f($X,$m,$Y,$n);
echo("\r\n".$X."\t".$Y."\t".trim(sprintf("%6.3f",$fXY)));
for($k=0;$k<count($c_i);$k++){
echo("\t".trim(sprintf("%6.3f",$ci[$k][0])));
echo("\t".trim(sprintf("%6.3f",$ci[$k][1]))); } } } }}
The est_N function computed the estimation errors or means or/and standard deviations of the mean for the one-variable functions (depending on X); the est_N2 function computed the estimation errors for two-variable functions (depending on X, Y):
function est_N($z,$a,&$c_i,$cif,$g,$ce){
 if($ce==1)echo("N\tX\tf(x)");else echo("N");
 for($i=0;$i<count($c_i);$i++)
echo("\t".$c_i[$i]);
 for($n=N_min;$n<N_max;$n++){
for($X=1;$X<p*$n;$X++){
$d=bino4($n,$X,10000);
if($ce==1) echo("\r\n".$n."\t".$X."\t".trim(sprintf("%6.3f",$cif($X,$n))));
for($i=0;$i<count($c_i);$i++){//pt fiecare ci
$ttt[$X][$i]=0;//esecuri
$conv=0;
foreach ($d as $XX => $nXX){//pt fiecare valoare posibila de esantion
if(out_ci_X($c_i[$i],$XX,$X,$n,$z,$a,$cif,$g)) $ttt[$X][$i] += $nXX;
$conv += $nXX; }
$ttt[$X][$i] *= 100/$conv;
if($ce==1)echo("\t".trim(sprintf("%3.3f",$ttt[$X][$i]))); } }
if ($ce > 1) est_MV($ttt,count($c_i),$n,p,0,$ce,$a); }}
function est_N2($z,$a,&$c_i,$f){
 echo("X\tY\tf(X,Y)");
 for($i=0;$i<count($c_i);$i++)
echo("\t".$c_i[$i]);
 for($n=N_min;$n<N_max;$n++){
$m = $n;
for($X=1;$X<p*$m;$X++){
$dX=bino4($m,$X,1000);
for($Y=1;$Y<p*$n;$Y++){
$dY=bino4($n,$Y,1000);
echo("\r\n".$X."\t".$Y."\t".trim(sprintf("%3.3f",$f($X,$m,$Y,$n))));
for($i=0;$i<count($c_i);$i++){
$ttt[$X][$Y][$i]=0;//esecuri
$conv=0;
foreach ($dX as $XX => $nXX){
foreach ($dY as $YY => $nYY){
if(out_ci_XY($c_i[$i],$XX,$X,$YY,$Y,$m,$n,$z,$a,$f))
$ttt[$X][$Y][$i] += $nXX*$nYY;
$conv += $nXX*$nYY; } }
echo("\t".trim(sprintf("%3.3f",100*$ttt[$X][$Y][$i]/$conv))); } } } }}
The est_MN function computed the means and/or the standard deviations of the mean for the two-variable functions (depending on X, Y):
function est_MN($z,$a,&$c_i,$f,$ce){
 echo("M\tN");
 for($i=0;$i<count($c_i);$i++)
echo("\t".$c_i[$i]);
for($m=N_min;$m<N_max;$m++){
for($n=N_min;$n<N_max;$n++){
for($i=0;$i<count($c_i);$i++){
if($ce % 2 == 0) $md[$i] = 0;
if($ce % 3 == 0) $va[$i] = 0; }
for($X=1;$X<p*$m;$X++){
$dX=bino4($m,$X,10000);
for($Y=1;$Y<p*$n;$Y++){
$dY=bino4($n,$Y,10000);
for($i=0;$i<count($c_i);$i++){
$ttt[$X][$Y][$i]=0;//esecuri
$conv=0;
foreach ($dX as $XX => $nXX){
foreach ($dY as $YY => $nYY){
if(out_ci_XY($c_i[$i],$XX,$X,$YY,$Y,$m,$n,$z,$a,$f))
$ttt[$X][$Y][$i] += $nXX*$nYY;
$conv += $nXX*$nYY; } }
if($ce % 2 == 0) $md[$i] += 100*$ttt[$X][$Y][$i]/$conv;
if($ce % 3 == 0) $va[$i] += pow(100*$ttt[$X][$Y][$i]/$conv-5,2); } } }
echo("\r\n".$m."\t".$n);
for($i=0;$i<count($c_i);$i++){
if($ce % 2 == 0)
echo("\t".trim(sprintf("%3.3f",$md[$i]/($m-1)/($n-1))));
if($ce % 3 == 0)
echo("\t".trim(sprintf("%3.3f",sqrt($va[$i]/(($m-1)*($n-1)-1))))); } } }}
The est_MV function computed the estimation errors means and/or standard deviations:
function est_MV(&$ttt,$n_c_i,$n,$p,$v,$ce,$a){
 for($i=0;$i<$n_c_i;$i++){
$md[$i]=$v; if($v)continue;
if($p==0.5){
for($X=1;$X<$p*$n;$X++) $md[$i]+=2*$ttt[$X][$i];
if(is_int($p*$n)) $md[$i]+=$ttt[$p*$n][$i];
$md[$i]/=($n-1);
} else {
for($X=1;$X<$p*$n;$X++) $md[$i]+=$ttt[$X][$i];
$md[$i]/=($n-1); } }
 if($ce % 3 == 0)
 for($i=0;$i<$n_c_i;$i++){
$ds[$i]=0;
if($p==0.5){
for($X=1;$X<$p*$n;$X++){
$ds[$i]+=2*pow($ttt[$X][$i]-$md[$i],2); }
if(is_int($p*$n)) $ds[$i]+=pow($ttt[$X][$i]-100*$a,2);//$md[$i],2);
$ds[$i]/=($n-2);//s
$ds[$i]=pow($ds[$i],0.5);
} else {
for($X=1;$X<$p*$n;$X++){
$ds[$i]+=pow($ttt[$X][$i]-100*$a,2);//$md[$i],2); }
$ds[$i]/=($n-2);//s
$ds[$i]=pow($ds[$i],0.5); } }
 if($ce % 2 == 0){
echo("\r\n".$n);
for($i=0;$i<$n_c_i;$i++)
echo("\t".trim(sprintf("%3.3f",$md[$i]))); }
 if($ce % 3 == 0){
echo("\r\n".$n);
for($i=0;$i<$n_c_i;$i++)
echo("\t".trim(sprintf("%3.3f",$ds[$i]))); }}
The est_R2 function compute and display, for a 100 random sample size and binomial variables values the experimental errors using the imposed (theoretical) value of significance level for all confidence interval calculation methods given also as parameter:
function est_R2($z,$a,&$c_i,$f){
 echo("p\tM\tN\tX\tY\tf(X,Y)");
 for($i=0;$i<count($c_i);$i++)
echo("\t".$c_i[$i]);
 for($k=0;$k<100;$k++){
$n = rand(4,1000);
$m = rand(4,1000);
$X = rand(1,$m-1);
$Y = rand(1,$n-1);
$dX=bino4($m,$X,10000);
$dY=bino4($n,$Y,10000);
echo("\r\n".$k."\t".$m."\t".$n."\t".$X."\t".$Y."\t".trim(sprintf("%3.3f",$f($X,$m,$Y,$n))));
for($i=0;$i<count($c_i);$i++){//pt fiecare ci
$ttt[$X][$Y][$i]=0;//esecuri
$conv=0;
foreach ($dX as $XX => $nXX){
foreach ($dY as $YY => $nYY){
if(out_ci_XY($c_i[$i],$XX,$X,$YY,$Y,$m,$n,$z,$a,$f))
$ttt[$X][$Y][$i] += $nXX*$nYY;
$conv += $nXX*$nYY;
}
}
echo("\t".trim(sprintf("%3.3f",100*$ttt[$X][$Y][$i]/$conv)));
}
 }
}
The est_C2 function compute and display for a fixed value of proportion X/n (usually the central point X/n = 1/2) the errors for a series of sample size (from N_min to N_max):
function est_C2($z,$a,&$c_i,$f){
 echo("X\tY\tf(X,Y)");
 for($i=0;$i<count($c_i);$i++)
echo("\t".$c_i[$i]);
 for($n=N_min;$n<N_max;$n+=4){
$m = $n;
//$X = (int)(3*$m/4); $Y = (int)($n/4);
$X = (int)($m/2); $Y = (int)($n/2);
$dX=bino4($m,$X,10000);
//$dY=bino4($n,$Y,10000);
$dY=$dX;
echo("\r\n".$X."\t".$Y."\t".trim(sprintf("%3.3f",$f($X,$m,$Y,$n))));
for($i=0;$i<count($c_i);$i++){
$ttt[$X][$Y][$i]=0;//esecuri
$conv=0;
foreach ($dX as $XX => $nXX){
foreach ($dY as $YY => $nYY){
if(out_ci_XY($c_i[$i],$XX,$X,$YY,$Y,$m,$n,$z,$a,$f))
$ttt[$X][$Y][$i] += $nXX*$nYY;
$conv += $nXX*$nYY;
}
}
echo("\t".trim(sprintf("%3.3f",100*$ttt[$X][$Y][$i]/$conv)));
}
 }
}
The ci function used in est_N, est_N2, est_X, and est_X2 functions compute the confidence interval for specified function with specified parameter and it was uses for the nine functions described in table 3. Corresponding to the functions described in table 3, were defines:

Binomial Distribution Sample Confidence Intervals Estimation 2. Proportion-like Medical Key Parameters
Sorana BOLBOACĂ^*, Andrei ACHIMAŞ CADARIU

 “Iuliu Haţieganu” University of Medicine and Pharmacy, Cluj-Napoca, Romania
* corresponding author, sbolboaca@umfcluj.ro


Abstract
The accuracy of confidence interval expressing is one of most important problems in medical statistics. In the paper was considers for accuracy analysis from literature, a number of confidence interval formulas for binomial proportions from the most used ones. For each method, based on theirs formulas, were implements in PHP an algorithm of computing. The performance of each method for different sample sizes and different values of binomial variable was compares using a set of criteria: the average of experimental errors, the standard deviation of errors, and the deviation relative to imposed significance level. A new method, named Binomial, based on Bayes and Jeffreys methods, was defines, implements, and tested. The results show that there is no method from considered ones, which to have a constant behavior in terms of average of the experimental errors, the standard deviation of errors, or the deviation relative to imposed significance level for every sample sizes, binomial variables, or proportions of them. More, this unfortunately behavior remains also for large and small subsets of natural numbers set. The Binomial method obtained systematically lowest deviations relative to imposed significance level (α = 5%) starting with a certain sample size.
Keywords
Confidence Intervals; Binomial Proportion; Proportion-like Medical Parameters

Introduction
In the medical studies, many parameters of interest are compute in order to assess the effect size of healthcare interventions based on 2X2 contingency table and categorical variables. If we look for example at diagnostic studies, we have the next parameters that can be express as a proportion: sensitivity, specificity, positive predictive value, negative predictive value, false negative rate, false positive rate, probability of disease, probability of positive test wrong, probability of negative test wrong, overall accuracy, probability of a negative test, probability of a positive test [1], [2], [3]. For the therapy studies experimental event rate, control event rate, absolute risk in treatment group, and absolute risk on control group are the proportion-like parameter [4], [5] which can be compute based on 2X2 contingency table. If we look at the risk factor studies, the individual risk on exposure group and individual risk on nonexposure group are the parameters of interest because they are also proportions [6]. Whatever the measure used, some assessment must be made of the trustworthiness or robustness of the findings, even if we talk about a p-value or confidence intervals. The confidences intervals are preferred in presenting of medical results because are relative close to the data, being on the same scale of measurements, while the p-values is a probabilistic abstraction [7]. Rothmans sustained that confidence intervals convey information about magnitude and precision of effect simultaneously, keeping these aspects of measurement closely linked [8]. The confidence intervals estimation for a binomial proportion is a subject debate in many scientific articles even if we talk about the standard methods or about non asymptotic methods [9], [10], [11], [12]. It is also known that the coverage probabilities of the standard interval can be erratically poor even if p is not near the boundaries 10, [13], [14], [15]. Some authors recommend that for the small sample size n (40 or less) to use the Wilson or the Jeffreys confidence intervals method and for the large sample size n (> 40) the Agresti-Coull confidence intervals method [10]. The aim of this research was to assess the performance for a binomial distribution samples of literature reported confidence intervals formulas using a set of criterions and to define, implement, and test the performance of a new original method, called Binomial.

Materials and Methods
In literature, we found some studies that present the confidence intervals estimation for a proportion based on normal probability distribution or on binomial probability distribution, or an binomial distribution and its approximation to normality [16]. In this study, we compared some confidence intervals expressions, which used the hypothesis of normal probability distribution, and some confidence intervals formula, which used the hypothesis of binomial probability distribution. It is necessary to remark that the assessment of the confidence intervals, which used the hypothesis of binomial distribution, is time-consuming comparing with the methods, which used the hypothesis of a normal distribution. In order to assess, both confidence intervals estimation aspects, quantitative and qualitative, and the performance of a confidence interval comparative with other methods we developed a module in PHP. The module was able to compute and evaluate the confidence intervals for a given X and a given n, build up a binomial distribution sample (XX) based on given X and compute the experimental errors of X values into its confidence intervals of xx from XX for each specified methods. We choose in our experiments a significance level equal with α = 5% (α = 0.05; noted a in our program) and the corresponding normal distribution percentile z_1-α/2 = 1.96 (noted z in the program) for the confidence intervals which used normal distribution probability. The value of sample size (n) was chosen in the conformity with the real number from the medical studies; in our experiments, we used 4 ≤ n ≤ 1000. The estimation of the errors were performed for the value of X variable which respect the rule 0 < X < n. The confidence intervals used in this study, with their references, are in table 1. The confidence intervals presented in table 1 were not the all methods used in confidence intervals estimation for a proportion but they are the methods most frequently reported in literature. Note that the performance of confidence interval calculation is dependent on method formula and eventually sample size and it must be independent on binomial variable value. Repetition of the experiment will and must produce same values of percentage errors for same binomial variable value, sample size, and method of calculation.

No	Methods	Lower and upper confidence limits
1	Wald [1]	; c = 0 (standard); c = 0.5 (continuity correction)
2	Wilson [10]	; Continuity correction:
3	ArcSine [17]	or 0 (X = 0) or 1 (X = n) (ArcS) Continuity correction methods: (ArcSC1) (ArcSC2) (ArcSC3)
4	Agresti-Coull [10]
5	Logit [10]	for 0 < X < n; Continuity correction:
6	Binomial distribution methods [10,17] Clopper-Pearson Jeffreys Bayesian, BayesianF Blyth-Still-Casella [18], [19]	for 0 < X < n we define the function: BinI(c1,c2,a) = BinS(c1,c2,a) = Clopper-Pearson: BinI(0,1,α/2), BinS(0,1,α/2) or 0 (X=0), 1(X=n) Jeffreys: BinI(0.5,0.5,α/2), BinS(0.5,0.5,α/2) BayesianF: BinI(1,1,α/2), BinS(1,1,α/2) or 0 (X=0), 1(X=n) Bayesian: BinI(1,1,α/2) or 0 (X=0) , α1/(n+1) (X=n), BinS(1,1,α/2) or 1- α1/(n+1) (X=0) , 1 (X=n) Blyth-Still-Casella: BinI(0,1,α1) or 0 (X=0), BinS(0,1,α2) or 1(X=n) where α1 + α2 = α and BinS(0,1,α2)-BinI(0,1,α1) = min.

Note: invCDF is invert of Fisher distribution
Table 1. Confidence intervals for proportion
In order to assess the methods used in confidence intervals estimations we implemented in PHP a matrix in which were memorized the name of the methods:
$c_i=array("Wald","WaldC","Wilson","WilsonC","ArcS","ArcSC1","ArcSC2", "ArcSC3",
"AC", "Logit", "LogitC", "BS", "Bayes" ,"BayesF" ,"Jeffreys" ,"CP");
where the WaldC was the Wald method with continuity correction, the WilsonC was the Wilson method with continuity correction, the AC was the Agresti-Coull method, the BS was the Blyth-Still method and the CP was the Clopper-Pearson method. For each formula presented in table 1 was created a PHP function which to be use in confidence intervals estimations. As it can be observes from the table 1, some confidence intervals functions (especially those based on hypothesis of binomial distribution) used the Fisher distribution.

Comparative Study of Monolayer Fatty Acids at the Air/Water Interface
Bianca Monica RUS Technical University of Cluj-Napoca, Romania

Abstract
The behavior of insoluble monolayer of surfactants spread at the air/water interfaces is frequently studied by recording compression isotherms.The compression isotherms given in terms of surface pressure versus mean molecular area are well reproducible for a chosen spreading molecular area of fatty acids, like stearic acid, oleic acid and linoleic acid. Mathematical modeling of these isotherms is performed by means of surface state equations. In the literature, a great variety of such equations have been proposed. In the present paper is made a study of the compression isotherms of oleic acid (OA) monolayer spread at the air /water interface on acidic aqueous solutions (pH = 2).
Keywords
Fatty Acid Monolayer, Equations of State, Air/water Interface

I. Introduction
The thermodynamic and structural characterization of fatty acid films at the air-water interface has received a remarkable attention over the past several decades [1-6]. The thermodynamic analysis of compression isotherms in terms of surface pressure (π, mN/m) versus mean molecular area (A, nm²/molecule) leads to equations of state that describes well the phase behavior of fatty acid monolayer. At very low surface pressures, monolayer might be considered to behave like two-dimensional gases. Therefore, in these conditions, the validity of the two-dimensional perfect gas state equation [4]:
πA = kT (1)
may be expected, where k is Boltzmann’s constant and T is the absolute temperature. Due to the experimental difficulties in measuring very low surface pressures, this region is less investigated. Much more data are available in the range of 0.2 mN/m < π <10 mN/m, where monolayer generally are in a liquid expanded state. Since the cross-sectional area of the surfactant molecules cannot be neglected and it represents a mean area necessity named also co-area, in equation (1) the A value has been corrected for a co-area (A₀), obtaining [9]:
π(A - A₀) = kT (2)
where A₀ is frequently taken as an adjusted parameter. Since in the liquid state the intermolecular interactions might be more important than the own molecular area of the surfactant, an internal pressure, π₀, can be introduced. Correcting π in equation (1) for this internal pressure, one obtains:
(π+π₀)A = kT (3)
By taking into account both intermolecular attractive forces and the above mentioned co-area, the state equation (1) becomes [10]:
(π +π₀)(A - A₀) = kT (4)
Further, a two-dimensional state equation, analogous with the Van der Waals equation:
(π + α/A²)(A - A₀) = kT (5)
has also been proposed. Equation (5) results from equation (4), by taking π₀ = α/A² on the base of the same reason, which in the case of a bulk gaseous phase leads to the presumption that the three-dimensional internal pressure is inversely proportional with the square of the volume. There were made several attempts to correlate the parameter α with molecular structure and with intermolecular interactions. On the basis of the scaled particle theory for fatty acids the state equation:
(π +α/A²)[A (1 – A₀'/A)²] = kT (6)
was derived [11]. In this equation A₀' = πd²/4 represents the area of a hard disc having its diameter equal to d, which is thought to be equivalent with the cross-sectional area of the saturated hydrocarbon chain. The correlation between A₀' and the co-area A₀ is A₀'=A₀/2. The parameter α was also correlated with molecular and intermolecular characteristics. For cohering uncharged films the equation:
(π + α/A³/2) (A - A₀) = kT (7)
was derived [12] and it was found to be in good agreement with some experimental results. According to the duplex film model, a surfactant monolayer spread at the air/water interface may be considered as a monolayer solution formed of the surfactant head groups dissolved in water and having the theoretical surface pressure π_h, covered by an oily layer consisting of the hydrophobic chains of surfactant molecules. Denoting by (-π₀) the spreading coefficient of this oil on the hypothetical surface solution of surfactant head groups and interfacial water, the overall surface pressure may be expressed as:
π = -π₀ + π_h (8)
By considering an insoluble monolayer, on the basis of the equality of the chemical potential of water in the bulk subphase and in the monolayer solution of headgroups, π_h may be given by a Butler type equation [13]:
(9)
where, A₁, a₁, γ₁ and x₁ stands for the molecular area, activity, activity coefficient and molar fraction of water molecules in the monolayer solution, respectively. Combination of equation (8) and (9) yields [14]:
(10)
By considering the headgroup monolayer to be a binary regular solution, one has
(11)
where β₁₂ and x₂ stands for the interaction parameter between the polar head group and water molecules in the monolayer and for the molar fraction of the head groups in the monolayer, respectively. Finally, the following equation of state is obtained:
(12)
It is well to mention that other state equations have also been derived based on statistical considerations of the conformation of hydrocarbon chain segments [15, 16]. In the present paper is made a study of the compression isotherms of oleic acid (OA) monolayer spread at the air /water interface on acidic aqueous solutions (pH = 2).

II. Material and Methods
In view of testing the state equations the OA monolayer has been chosen as model system. The compression isotherm recorded on a subphase with pH = 2 has been used since at this pH the protolytic equilibrium in the monolayer is shifted practically completely towards the neutral OA molecules [7]. The experimental π vs. A curve is given in fig.1 (curve 1). The theoretical curve, corresponding to equation (1) is visualized in the same figure (curve 2). As seen, at high A values important deviations from the perfect behavior appear in the negative direction, indicating the role of the intermolecular attractive forces. On the contrary, at lower A values, the deviations become positive.

III. Results and Discussion
The other state equations contain single (equations (2) and (3) or even two equations 4-7) parameters to be derived from experimental data. Since a good mathematical description of the compression isotherms by means of state equations can be expected in the case of expanded films, i.e. at low surface pressures, the π ≤ 7 mN/m has been chosen. This domain is delimited in Fig. 1 by dashed straight lines, and we had in this region a number of 17 experimental points, i.e. Ai - πi pairs. These data were processed following a curve fitting method. For this purpose, the standard deviation □ of the experimental points from the theoretical ones was calculated and minimized by performing a systematic variation of the parameter / parameters to be derived.

Fig. 1. πvs. A isotherms for OA monolayer. Experimental curve(1); theoretical curves (2-5) calculated by using parameters given in Table 1 and state equations: curve (2) – equation (1); (3) - equation (2); (4) - equations (3); (5) - equation (4).
In the case of equation (2) and (3), by varying A₀ and π₀, respectively, the minimum standard deviation, _m, was determined, which is given in Table 1, together with the corresponding parameter value. By using the latters the theoretical ones, given in Fig. 1, were calculated (curves 3 and 4). As seen, equation (2) gives a rather unrealistic, negative A₀ value, while equation (3) yields a π₀ value corresponding to intermolecular attraction. Neither equation (2), nor equation (3) is able to give a good description of the experimental curve, the _m value being rather high. By using equations with two adjustable parameters, a double minimization of □ is to be performed. To this end, one of the parameters is maintained at a constant value, and by varying the other one, the minimum of □, denoted as □_m is determined. The systematic variation of the first parameter allows us to determine the minimum of the □_m values, (□_m)_m, indicating the best values of both parameters to be derived. These double minimum values of ∆ are given in Table 1, and the theoretical curve calculated by means of equation (4), using π₀ and A₀ values given in Table 1, is also visualized in Fig. 1, curve (5). Obviously, the use of two adjustable parameters entails a spectacular improvement of the approximation. Table 1. Parameters of the state equation and standard deviations derived for OA monolayer (π ≤ 7 mN/m)

Eq	π0 (mN/m)	A0 (nm2/molec)	α*	β12∙1020 (Nm)	□** (mN/M)
(1)	-	-	-	-	6.09
(2)	-	-1.123	-	-	1.95
(3)	6.0	-	-	-	1.19
(4)	10.5	0.169	-	-	0.29
(6)	-	0.268	5.691	-	0.17
(7)	10.46	-	-	-0.08	0.29
(9)	-	-	0.842	-1.07	0.15
(10)	-	-	8.601	-2.28	0.09
(11)	-	-	1.153	-5.73	0.13

*Units: 1) 10-30 Nm2; 2) 10 -20 Nm; 3) 10-38 Nm2; in all cases per molecule
**It means _m in the case of equations (2) and (3), and (_m)_m with equations (4), (6), (7) and (9-11). As seen from Table 1, equation (6) gives better results than the others. As far as equation (7) is concerned, its use raises several problems, related to calculation of the molar fractions in the monolayer.

References
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8. Tomoaia-Cotisel M., Mocanu A., Pop V.D, Plesa D., Plesa S., Albu I., Equations of State for Films of Fatty Acids at the Air/Water Interface, The VII^th Symposium of Colloid and Surface Chemistry, Bucharest, 2002.
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13. Butler J.A.V., Proc. Royal. Soc. London, Ser. A, 135, 1932, p. 348.
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