Unit 1: Analysis of Simple Comparative Experiments

1 Unpaired Design

Example of Comparing Modified and Unmodified Cement

1.1 Input Data for Analysis

mod <- c(16.85, 16.40, 17.21, 16.35, 16.52, 17.04, 16.96, 17.15, 16.59, 16.57)
unmod <- c(17.50, 17.63, 18.25, 18.00, 17.86, 17.75, 18.22, 17.90, 17.96, 18.15)

1.2 Draw boxplot

boxplot(mod,unmod)

1.3 Perform two sample t-test (with equal variance assumption)

t.test(mod,unmod,var.equal=TRUE)

## 
##  Two Sample t-test
## 
## data:  mod and unmod
## t = -9.1094, df = 18, p-value = 3.678e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.4250734 -0.8909266
## sample estimates:
## mean of x mean of y 
##    16.764    17.922

t.test(mod,unmod) #(Welch approximation)

## 
##  Welch Two Sample t-test
## 
## data:  mod and unmod
## t = -9.1094, df = 17.025, p-value = 5.894e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.4261741 -0.8898259
## sample estimates:
## mean of x mean of y 
##    16.764    17.922

1.4 Checking normality assumptions

# normality check
qqnorm(mod)
qqline(mod,col=2)

qqnorm(unmod)
qqline(unmod,col=4)

1.5 Testing equality of variance

var.test(mod,unmod)

## 
##  F test to compare two variances
## 
## data:  mod and unmod
## F = 1.6293, num df = 9, denom df = 9, p-value = 0.4785
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.4046845 6.5593806
## sample estimates:
## ratio of variances 
##           1.629257

# alternatively we can using these two steps
f <- var (mod)/var (unmod);f

## [1] 1.629257

pv <- 2* pf (f, df1 = 9, df2=9, lower.tail = F); pv

## [1] 0.4784603

2 Paired Design

Consider an experiment to compare two methods, MSI and SIB, to determine chlorine content in sewage effluents. Eight samples were collected at different doses and contact times. Two methods were applied to each of the eight samples. It is a paired comparison design because the pair of treatments are applied to the same samples (or units).

2.1 Read Data

chlorine <- read.table ("data/chlorine.txt", header = T)

chlorine

Sample	MSI	SIB	Diff
1	0.39	0.36	-0.03
2	0.84	1.35	0.51
3	1.76	2.56	0.80
4	3.35	3.92	0.57
5	4.69	5.35	0.66
6	7.70	8.33	0.63
7	10.52	10.70	0.18
8	10.92	10.91	-0.01

MSI <- chlorine$MSI
SIB <- chlorine$SIB
Sample <- chlorine$Sample

2.2 Drawing boxplot

boxplot (MSI, SIB)

2.3 Drawing matplot

matplot(chlorine[,c("MSI","SIB")],
        type = "b", xlab = "Sample", ylab = "Chlorine")

2.4 Drawing difference vs sample

plot (SIB - MSI ~ Sample, type = "h")
abline (h = 0)

2.5 Testing for the difference in means

t.test (MSI, SIB)

## 
##  Welch Two Sample t-test
## 
## data:  MSI and SIB
## t = -0.19831, df = 13.993, p-value = 0.8457
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -4.888792  4.061292
## sample estimates:
## mean of x mean of y 
##   5.02125   5.43500

t.test (MSI, SIB, paired = T)

## 
##  Paired t-test
## 
## data:  MSI and SIB
## t = -3.6454, df = 7, p-value = 0.008228
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.6821314 -0.1453686
## sample estimates:
## mean of the differences 
##                -0.41375

t.test (MSI-SIB)

## 
##  One Sample t-test
## 
## data:  MSI - SIB
## t = -3.6454, df = 7, p-value = 0.008228
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.6821314 -0.1453686
## sample estimates:
## mean of x 
##  -0.41375