1 Introduction

This is the data analysis example E from Cox and Snell’s Applied Statistics. This is a before and after example for blood pressure. The tablulated data gives the supine systolic and diastolic for 15 patients with moderate hypertension before and after taking 25mg of Captopril.

MacGregor, Markandu, Roulston and Jones (1979)

Patient <- c(1:15)
bS <- c(210,169,187,160,167,176,185,206,173,146,174,201,198,148,154)
aS <- c(201,165,166,157,147,145,168,180,147,136,151,168,179,129,131)
dS <- aS - bS
bD <- c(130,122,124,104,112,101,121,124,115,102,98,119,106,107,100)
aD <- c(125,121,121,106,101,85,98,105,103,98,90,98,110,103,82)
dD <- aD - bD
data <- data.frame(Patient,bS,aS,dS,bD,aD,dD)

data1 <- data
colnames(data) <- c("Patient No.", "before", "after", "difference", "before", "after", "difference")

data %>%
  kbl(caption = "Blood pressures (mm Hg) before and after Captopril") %>%
  kable_classic(full_width =  T, html_font= "Cambria") %>%
  add_header_above(c(" " = 1, "Systolic" = 3, "Diastolic" =3))
Blood pressures (mm Hg) before and after Captopril
Systolic
Diastolic
Patient No.  before after difference before after difference
1 210 201 -9 130 125 -5
2 169 165 -4 122 121 -1
3 187 166 -21 124 121 -3
4 160 157 -3 104 106 2
5 167 147 -20 112 101 -11
6 176 145 -31 101 85 -16
7 185 168 -17 121 98 -23
8 206 180 -26 124 105 -19
9 173 147 -26 115 103 -12
10 146 136 -10 102 98 -4
11 174 151 -23 98 90 -8
12 201 168 -33 119 98 -21
13 198 179 -19 106 110 4
14 148 129 -19 107 103 -4
15 154 131 -23 100 82 -18

In order to eliminate the Placebo Effect you should actually carry out a randomised controlled trial where you are not recording before against after but placebo vs Captopril. Then you randomly assign patients to each of the two treatments and have a period between the treatments where their natural levels can be restored. In that case as the treatment would be assigned randomly the ordering of placebo and drug would be random between patients.

The conditions for eliminating the Placebo Effect in this case are the stability of the measurements before and after treatment which makes a Placebo Effect less likely but not impossible.

These are paired data and so the obvious comparisons are of the difference between the before and after measurements. These can be between the current measures or transformed versions of the measures if considered appropriate. In this case transformationa does not seem to be required. All of the differences in systolic blood pressure are negative and this infers a clear relationship between the drug and lowering systolic blood pressure with a significant effect size.

However there could be a more complex relationship between the blood pressure before and after treatment. The drug might be more effective for certain initial levels and less effective at others. To test this you need to plot the difference against the initial value.

1.1 Plotting the Effect Against the Initial Values

ggplot() +
  geom_point(data=data1, aes(bS,dS))+
  stat_smooth(method = "lm", col ="red")+
  labs(y ="Difference in systolic pressure (mmHg)",x ="Initial systolic pressure (mmHg)", title="Plot of the Effect Size Against the Intitial Measure")

In this case there is no obvious relationship between the effect and the initial measurement.

1.2 The Paired T-test

t.test(bS,aS, paired=TRUE)
## 
##  Paired t-test
## 
## data:  bS and aS
## t = 8.1228, df = 14, p-value = 1.146e-06
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  13.93409 23.93258
## sample estimates:
## mean of the differences 
##                18.93333
t.test(bD,aD, paired=TRUE)
## 
##  Paired t-test
## 
## data:  bD and aD
## t = 4.1662, df = 14, p-value = 0.0009511
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##   4.496118 14.037215
## sample estimates:
## mean of the differences 
##                9.266667