Difference between revisions of "Team:NCKU Tainan/Model Statistics Analysis"

Line 214: Line 214:
 
                     <p>(Q-Q plot)<img src="/wiki/images/2/2b/T--NCKU_Tainan--project-modeling-statistic-image6.jpg"></p>
 
                     <p>(Q-Q plot)<img src="/wiki/images/2/2b/T--NCKU_Tainan--project-modeling-statistic-image6.jpg"></p>
 
                     <p>This model conforms the assumption and R-square is equal to 0.9541, so we think the cubic polynomial regression model is suitable.</p>  
 
                     <p>This model conforms the assumption and R-square is equal to 0.9541, so we think the cubic polynomial regression model is suitable.</p>  
                     <p>(prediction interval)<br>Formula: \(\bar{X}\pm t_{\frac{\alpha}{2},n-1}S\sqrt{1+\frac{1}{n}}\)<img src="/wiki/images/5/5e/T--NCKU_Tainan--project-modeling-statistic-image7.jpg"></p>
+
                     <p>(prediction interval)<br>Formula: \(\bar{X}\pm t_{\frac{\alpha}{2},n-1}S\sqrt{1+\frac{1}{n}}\)</p>
 +
<img src="/wiki/images/5/5e/T--NCKU_Tainan--project-modeling-statistic-image7.jpg">
 
                     <p>And look the following, the procedure is same in the 5mM/L and 120mM/L.</p><p class="center">(5mM/L)<img src="/wiki/images/b/b4/T--NCKU_Tainan--project-modeling-statistic-image8.jpg"><img src="/wiki/images/6/69/T--NCKU_Tainan--project-modeling-statistic-image9.jpg"><img src="/wiki/images/c/c9/T--NCKU_Tainan--project-modeling-statistic-image10.jpg"><img src="/wiki/images/b/b6/T--NCKU_Tainan--project-modeling-statistic-image11.jpg"><img src="/wiki/images/f/ff/T--NCKU_Tainan--project-modeling-statistic-image12.jpg">(120mM/L)<img src="/wiki/images/8/84/T--NCKU_Tainan--project-modeling-statistic-image13.jpg"><img src="/wiki/images/a/a5/T--NCKU_Tainan--project-modeling-statistic-image14.jpg"><img src="/wiki/images/0/09/T--NCKU_Tainan--project-modeling-statistic-image15.jpg"><img src="/wiki/images/e/ef/T--NCKU_Tainan--project-modeling-statistic-image16.jpg"><img src="/wiki/images/f/f1/T--NCKU_Tainan--project-modeling-statistic-image17.jpg"></p>
 
                     <p>And look the following, the procedure is same in the 5mM/L and 120mM/L.</p><p class="center">(5mM/L)<img src="/wiki/images/b/b4/T--NCKU_Tainan--project-modeling-statistic-image8.jpg"><img src="/wiki/images/6/69/T--NCKU_Tainan--project-modeling-statistic-image9.jpg"><img src="/wiki/images/c/c9/T--NCKU_Tainan--project-modeling-statistic-image10.jpg"><img src="/wiki/images/b/b6/T--NCKU_Tainan--project-modeling-statistic-image11.jpg"><img src="/wiki/images/f/ff/T--NCKU_Tainan--project-modeling-statistic-image12.jpg">(120mM/L)<img src="/wiki/images/8/84/T--NCKU_Tainan--project-modeling-statistic-image13.jpg"><img src="/wiki/images/a/a5/T--NCKU_Tainan--project-modeling-statistic-image14.jpg"><img src="/wiki/images/0/09/T--NCKU_Tainan--project-modeling-statistic-image15.jpg"><img src="/wiki/images/e/ef/T--NCKU_Tainan--project-modeling-statistic-image16.jpg"><img src="/wiki/images/f/f1/T--NCKU_Tainan--project-modeling-statistic-image17.jpg"></p>
                     <p>2. According to the previous part, we can say that 0.1mM/L and 5mM/L are difference group by the statistical significance. In order to precisely distinguish these two groups, choose the time at which the 5mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.<img src="/wiki/images/e/ec/T--NCKU_Tainan--project-modeling-statistic-image18.jpg"></p>
+
                     <p>2. According to the previous part, we can say that 0.1mM/L and 5mM/L are difference group by the statistical significance. In order to precisely distinguish these two groups, choose the time at which the 5mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.</p><img src="/wiki/images/e/ec/T--NCKU_Tainan--project-modeling-statistic-image18.jpg">
 
                     <p>By the above tables, we can see that 5mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 101 mins. Hence, it can be expected to have a higher accuracy to take the data after 101 mins.</p>
 
                     <p>By the above tables, we can see that 5mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 101 mins. Hence, it can be expected to have a higher accuracy to take the data after 101 mins.</p>
 
                     <br>
 
                     <br>
                     <p>(prediction interval of 0.1 &amp;5 mM/L)<img src="/wiki/images/c/c0/T--NCKU_Tainan--project-modeling-statistic-image1.jpg"></p>
+
                     <p>(prediction interval of 0.1 &amp;5 mM/L)</p>
                     <p>3. Same with procedure 2, choose the time at which the 120mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.<img src="/wiki/images/b/b9/T--NCKU_Tainan--project-modeling-statistic-image19.jpg"></p>
+
<img src="/wiki/images/c/c0/T--NCKU_Tainan--project-modeling-statistic-image1.jpg">
 +
                     <p>3. Same with procedure 2, choose the time at which the 120mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.</p>
 +
<img src="/wiki/images/b/b9/T--NCKU_Tainan--project-modeling-statistic-image19.jpg">
 
                     <p>By the above tables, we can find that 120mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 88 mins. Hence, it can be expected to have a higher accuracy to take the data in 88 mins.</p>
 
                     <p>By the above tables, we can find that 120mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 88 mins. Hence, it can be expected to have a higher accuracy to take the data in 88 mins.</p>
 
                     <br>
 
                     <br>
                     <p>(prediction interval of 0.1 &amp;120 mM/L)<img src="/wiki/images/0/04/T--NCKU_Tainan--project-modeling-statistic-image2.jpg"></p>
+
                     <p>(prediction interval of 0.1 &amp;120 mM/L)</p>
                     <h1 class="center">Appendix</h1>
+
<img src="/wiki/images/0/04/T--NCKU_Tainan--project-modeling-statistic-image2.jpg">
 +
                     <h1 class="title-line" id="appendix">Appendix</h1>
 
                     <pre><code class="R">#0.1 vs 5
 
                     <pre><code class="R">#0.1 vs 5
 
#0.1 urine plot
 
#0.1 urine plot
Line 313: Line 317:
 
                         <li><a href="#" onclick="return toEvent('pair');">Paired-difference T test</a></li>
 
                         <li><a href="#" onclick="return toEvent('pair');">Paired-difference T test</a></li>
 
                         <li><a href="#" onclick="return toEvent('reg');">Regression &amp; Prediction</a></li>
 
                         <li><a href="#" onclick="return toEvent('reg');">Regression &amp; Prediction</a></li>
 +
<li><a href="#appendix" onclick="return toEvent('appendix');">Appendix</a></li>
 
                       </ul>
 
                       </ul>
 
                     </div>
 
                     </div>

Revision as of 03:37, 11 October 2016

Project Modeling : Statistics Analysis - iGEM NCKU

PROJECT / Model / Statistics Analysis
Model: Statistics Analysis
Introduction

In medicine, the blood glucose exceeding 5mM/L implies having diabetes, and it exceeding 120mM/L implies being a severe patient. Finding the person whose glucose concentration is over 5mM/L or 120mM/L is our target. First, we prove that there has difference between 0.1mM/L and 5mM/L (120mM/L) in the paired-difference T test part. And, we use the regression and prediction intervals to distinguish exceeding 120mM/L and 5mM/L from exceeding 0.1mM/L. From the result, 5mM/L can be distinguished from 0.1mM/L after 101 minutes, and 120mM/L can be distinguished from 0.1mM/L after 88 minutes.

Paired-difference T test

In medicine, the blood glucose exceeding 5mM/L implies having diabetes, and it exceeding 120mM/L implies being a severe patient. Hence, we use paired-difference test to analyze whether the difference of these two groups (0.1 vs 5mM/L or 0.1 vs 120mM/L) have statistical significance in this part.



Procedure


1. Guessing that there is difference over 90 minutes, we analyze the data of 0.1mM/L and 5mM/L at 90 minutes first.

(Data)

Experiment


Types
First Second ... Twelfth
0.1mM/L 358 368 ... 338
5mM/L 369 386 ... 385

Because the experiments of 0.1mM/L and 5mM/L have correlation and they are small sample, we choose the paired-difference test to examine whether there is difference in two groups at 90 minutes.


Analysis:
(use one-tailed test)

\(H_0:\mu_0\le0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ H_\alpha:\mu_0\ge0\)

Because the sample is small, we choose the t distribution to test.

d(=5mM/L-0.1mM/L) 11 18 ... 47

\(n\) (Number of paired differences)=12
\(\bar{d}\) (Mean of the sample differences)= 26.16666667
\(s_d\) (Standard deviation of the sample differences) = \(\sqrt{\frac{\sum(d_i-\bar{d})^2}{n-1}}\) = 13.19664788

Test statistic t = \(\frac{\bar{d}-0}{s_d/\sqrt{n}}\) = 6.868713411 \(\gt t_{0.05,11}\) = 1.795885
P(t\(\gt\)6.868713411)= 1.995084e-05
Hence, We conclude that there is a difference between 0.1 and 5mM/L.

2. Find the minimal time at which there is statistical significance by paired-difference test.(use R program & α=0.05)

(Data)

Types

Time(min)
0.1mM/L ... 0.1mM/L 5mM/L ... 5mM/L
0 295 ... 276 292 ... 291
1 296 ... 276 290 ... 291
2 301 ... 277 292 ... 287
.
.
.
.
.
.
... .
.
.
.
.
.
... .
.
.
119 392 ... 372 420 ... 443
120 392 ... 374 427 ... 439

By using R program
\(t_{0.05,11}\) = 1.795885

statistic
Time(min)

T statistic (\(t_0\))

0 2.283227
1 1.992688
.
.
.
.
.
.
119 11.908155
120 15.538544

Hence, we can find \(t_0 \gt t_{0.05,11}\) at every time.

data0.1<-read.csv("C:/Users/Rick/Desktop/NCKUactivity/iGEM/819test/0.1urine.csv",header=T)
data5<-read.csv("C:/Users/Rick/Desktop/ NCKUactivity/iGEM/819test/5urine.csv",header=T)
difference<-cbind(data5[,2]-data0.1[,2],data5[,3]-data0.1[,3],data5[,4]-data0.1[,4],data5[,5]-data0.1[,5],data5[,6]-data0.1[,6],data5[,7]-data0.1[,7],data5[,8]-data0.1[,8],data5[,9]-data0.1[,9],data5[,10]-data0.1[,10],data5[,11]-data0.1[,11],data5[,12]-data0.1[,12],data5[,13]-data0.1[,13])
dbar<-difference%*% rep(1/12,12)
var<- ((difference -rep(dbar, 12))^2%*%rep(1,12))/11
sd<-var^0.5
t<-(dbar/sd)*(12^0.5)

3. Same with procedure 2, find the minimal time at which there is statistical significance by paired-difference test. (use R program, α=0.05, remove a outlier data)

By using R program
\(t_{0.05,10}\) = 1.812461

statistic
Time(min)

T statistic (\(t_0\))

0 4.720629
1 4.580426
.
.
.
.
.
.
119 12.428324
120 14.240245

Hence, we can find \(t_0 \gt t_{0.05,10}\) at every time.

Conclusion:
        We can say there are the difference groups (0.1 vs 5 mM/L or 0.1 vs 120mM/L) in part 1 with the data. And then, we need to find an accurate value which can distinguish between two groups.

data0.1<-read.csv("C:/Users/Rick/Desktop/ NCKUactivity/iGEM/819test/0.1urine.csv",header=T)
data120<-read.csv("C:/Users/Rick/Desktop/ NCKUactivity/iGEM/819test/120urine.csv",header=T)
difference<-cbind(data120[,2]-data0.1[,2],data120[,3]-data0.1[,3],data120[,4]-data0.1[,4],data120[,5]-data0.1[,5],data120[,6]-data0.1[,6],data120[,8]-data0.1[,8],data120[,9]-data0.1[,9],data120[,10]-data0.1[,10],data120[,11]-data0.1[,11],data120[,12]-data0.1[,12],data120[,13]-data0.1[,13])
dbar<-difference%*% rep(1/11,11)
var<- ((difference -rep(dbar, 11))^2%*%rep(1,11))/10
sd<-var^0.5
t<-(dbar/sd)*(11^0.5)
Regression & Prediction interval

        Our product uses the fluorescence intensity to obtain the concentration of blood glucose. Therefore, whether we could precisely distinguish exceeding 120mM/L and 5mM/L from exceeding 0.1mM/L becomes the most important problem. We use the skill of the prediction interval in this part.


Procedure

1. Use the regression to find the model of 0.1, 5, 120mM/L, and show the prediction interval plot.

(0.1mM/L)

(summary)

(residual plot)

(Q-Q plot)

This model conforms the assumption and R-square is equal to 0.9541, so we think the cubic polynomial regression model is suitable.

(prediction interval)
Formula: \(\bar{X}\pm t_{\frac{\alpha}{2},n-1}S\sqrt{1+\frac{1}{n}}\)

And look the following, the procedure is same in the 5mM/L and 120mM/L.

(5mM/L)(120mM/L)

2. According to the previous part, we can say that 0.1mM/L and 5mM/L are difference group by the statistical significance. In order to precisely distinguish these two groups, choose the time at which the 5mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.

By the above tables, we can see that 5mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 101 mins. Hence, it can be expected to have a higher accuracy to take the data after 101 mins.


(prediction interval of 0.1 &5 mM/L)

3. Same with procedure 2, choose the time at which the 120mmo/L and 0.1mmo/L prediction interval started to diverge to be the recommended prediction time.

By the above tables, we can find that 120mM/L low prediction line exceeds 0.1mM/L up prediction line when the time is more than 88 mins. Hence, it can be expected to have a higher accuracy to take the data in 88 mins.


(prediction interval of 0.1 &120 mM/L)

Appendix

#0.1 vs 5
#0.1 urine plot
data<-read.csv("C:/Users/Rick/Desktop/NCKUactivity/iGEM/819test/0.1urine.csv",header=T)
x1<-rep(data[,1],12)
x2<-x1^2
x3<-x1^3
y1<-c(data[,2],data[,3],data[,4],data[,5],data[,6],data[,7], data[,8],data[,9],data[,10], data[,11],data[,12],data[,13])
model<-lm(y1~x1+x2+x3)
yhat0.1<-model$fit
plot(x1,y1,ylim=c(250,450),xlab= "time(mins) ",ylab= "Fluorescent", cex.lab=1.5)
#lines(data[,1], yhat0.1[1:121],lwd=3,col=2)
n=121*12 
mse<-sum((y1-yhat0.1)^2)/(n-3)  # calculate MS_Res(=σ^2)
X<-cbind(1,x1,x2,x3)
se<-sqrt(mse*(1+X[1,]%*%solve(t(X)%*%X)%*%X[1,]))
up0.1<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.975,df=n-2)*se
low0.1<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.025,df=n-2)*se
lines(data[,1], up0.1[1:121], col=2,lwd=3)
lines(data[,1], low0.1[1:121],col=2,lwd=3)
#5 urine plot
data<-read.csv("C:/Users/Rick/Desktop/NCKUactivity/iGEM/819test/5urine.csv",header=T)
x1<-rep(data[,1],12)
x2<-x1^2
x3<-x1^3
y1<-c(data[,2],data[,3],data[,4],data[,5],data[,6],data[,7], data[,8],data[,9],data[,10], data[,11],data[,12],data[,13])
model<-lm(y1~x1+x2+x3)
yhat5<-model$fit 
par(new=TRUE)
plot(x1,y1,ylim=c(250,450),xlab= "time(mins) ",ylab= "Fluorescent", cex.lab=1.5)
#lines(data[,1], yhat5[1:121],lwd=3,col=2)
n=121*12 
mse<-sum((y1-yhat5)^2)/(n-3)  # calculate MS_Res(=σ^2)
X<-cbind(1,x1,x2,x3)
se<-sqrt(mse*(1+X[1,]%*%solve(t(X)%*%X)%*%X[1,]))
up5<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.975,df=n-2)*se
low5<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4] +qt(.025,df=n-2)*se
lines(data[,1], up5[1:121], col=3,lwd=3)
lines(data[,1], low5[1:121], col=3,lwd=3)
title(main="0.1 vs 5mM/L",cex.main=4)
Name<-c(expression(paste("0.1mM/L")),expression(paste("5mM/L")))
legend("topleft", Name, ncol = 1, cex = 1.5, col=c("red","green"),lwd = c(2,2), bg = 'gray90')
#0.1 vs 120
#0.1 urine plot
data<-read.csv("C:/Users/Rick/Desktop/NCKUactivity/iGEM/819test/0.1urine.csv",header=T)
x1<-rep(data[,1],12)
x2<-x1^2
x3<-x1^3
y1<-c(data[,2],data[,3],data[,4],data[,5],data[,6],data[,7], data[,8],data[,9],data[,10], data[,11],data[,12],data[,13])
model<-lm(y1~x1+x2+x3)
yhat0.1<-model$fit
plot(x1,y1,ylim=c(250,450),xlab= "time(mins) ",ylab= "Fluorescent", cex.lab=1.5)
#lines(data[,1], yhat0.1[1:121],lwd=3,col=2)
n=121*12 
mse<-sum((y1-yhat0.1)^2)/(n-3)  # calculate MS_Res(=σ^2)
X<-cbind(1,x1,x2,x3)
se<-sqrt(mse*(1+X[1,]%*%solve(t(X)%*%X)%*%X[1,]))
up0.1<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.975,df=n-2)*se
low0.1<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficiens[4]+qt(.025,df=n-2)*se
lines(data[,1], up0.1[1:121], col=2,lwd=3)
lines(data[,1], low0.1[1:121],col=2,lwd=3)
#120 urine plot
data<-read.csv("C:/Users/Rick/Desktop/NCKUactivity/iGEM/819test/120urine.csv",header=T)
x1<-rep(data[,1],10)
x2<-x1^2
x3<-x1^3
y1<-c(data[,2],data[,3],data[,4],data[,5],data[,6],data[,9],data[,10],data[,11],data[,12],data[,13])
model<-lm(y1~x1+x2+x3)
yhat120<-model$fit 
par(new=TRUE)
plot(x1,y1,ylim=c(250,450),xlab= "time(mins) ",ylab= "Fluorescent", cex.lab=1.5)
#lines(data[,1], yhat120[1:121],lwd=3,col=2)
n=121*10 
mse<-sum((y1-yhat120)^2)/(n-3)  # calculate MS_Res(=σ^2)
X<-cbind(1,x1,x2,x3)
se<-sqrt(mse*(1+X[1,]%*%solve(t(X)%*%X)%*%X[1,]))
up120<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.975,df=n-2)*se
low120<-model$coefficients[1]+x1*model$coefficients[2]+x2*model$coefficients[3]+x3*model$coefficients[4]+qt(.025,df=n-2)*se
lines(data[,1], up120[1:121], col=4,lwd=3)
lines(data[,1], low120[1:121], col=4,lwd=3)
title(main="0.1 vs 120mM/L",cex.main=4)
Name<-c(expression(paste("0.1mM/L")),expression(paste("120mM/L")))
legend("topleft", Name, ncol = 1, cex = 1.5, col=c("red","green"),lwd = c(2,2), bg = 'gray90')