# Paired t-test or repeated measurement

I am designing RCT in the same patient (treatment on one side of the face and control on the contralateral side of the face)

The RCT will be 1 month long with measurements happening at 2 and 4 week postoperatively.

I am interested in differences between treatment and control sides at each time follow up.

I am not sure whether a simple paired t-test for or a repeated measure ANOVA would be more appropriate.

Thankyou

• You could do a paired t-test comparing wks 2 & 4 on treated side, another comparing wks 2 & 4 on other side, and then add'l paired tests for your comparisons of interest. (Use Wilcoxon signed-rank test, if data not normal) // A more elegant and perhaps more informative approach would be to use an ANOVA design with fixed effects Time (wks 2 & 4), Procedure (treatment and control side); Subjects as a random effect. For the ANOVA you'd need to check that variances for 2 & 4 don't differ markedly and vars for Treat & Ctrl don't differ markedly. Also check residuals for normality. Then contrasts. Sep 30 '18 at 7:00

I would want to look at the main comparisons of interest first, to see if I had interesting results. Then explore further with additional paired tests or with ad hoc comparisons based on an ANOVA. Shown below are some procedures in R (mainly for paired tests) that may be useful as you start your analysis with (real) data.

Suppose you have 20 patients with (fake) measurements as shown below:

      sbj    c2    c4    t2    t4
[1,]   1 19.11 15.56 13.66 17.58
[2,]   2  9.02 17.30 15.50 17.05
[3,]   3 12.38 13.78 19.30 20.92
[4,]   4 11.56 14.87 18.61 16.37
[5,]   5  6.34  9.21 13.31 14.68
[6,]   6 11.65 10.51 15.55 18.58
[7,]   7 12.05  8.89 11.59 12.52
[8,]   8  8.11  8.99 13.46 12.04
[9,]   9 10.69 12.03 17.96 15.68
[10,]  10 18.36 16.33 23.43 25.72
[11,]  11 12.04 11.16 11.76 16.80
[12,]  12  8.13  3.43  9.06 11.35
[13,]  13 17.74 15.41 22.37 20.41
[14,]  14  7.60  8.75 10.49 15.30
[15,]  15 11.97  8.79 13.54 18.97
[16,]  16 14.54 14.55 14.83 20.07
[17,]  17  9.82 12.43 13.22 16.61
[18,]  18  8.07 11.12 15.50 16.04
[19,]  19 16.24 17.27 18.72 20.71
[20,]  20 14.36 17.75 17.02 22.98


Is there a significant difference between the treatment and control sides at Week 2, and if so, is this difference large enough to be of importance? The summary below shows a difference of about 3.5 and the stripchart shows mainly positive differences.

summary(t2 - c2)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-5.450   1.410   3.650   3.455   6.590   7.430


A Shapiro-Wilk normality test shows that the differences are consistent with sampling from a normal distribution.

shapiro.test(t2-c2)

Shapiro-Wilk normality test

data:  t2 - c2
W = 0.91938, p-value = 0.09639


A paired t test shows that this difference is highly significant [P-value about 0.0002, and confidence interval $$(1.88, 5.03)$$]. Its clinical importance, not ordinarily a statistical issue, would be judged by people who understand what the measurements actually mean in terms of patient outcomes.

t.test(t2 - c2)     # alternatively 't.test(t2, c2, paired=T)'

One Sample t-test

data:  t2 - c2
t = 4.5933, df = 19, p-value = 0.0001985
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
1.880662 5.029338
sample estimates:
mean of x
3.455


The corresponding comparison at Week 4 shows an even larger difference. A Shapiro-Wilk test [not shown] does not reject normality. A paired t test [not shown] returns p-value = 4.668e-08.

summary(t4 - c4)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-0.250   3.583   5.115   5.112   6.697  10.180


Exploring further, if these were real data, it might be of interest that the control measurements showed no significant difference between weeks 2 and 4, while the treatment measurements show a highly significant difference. So the large difference between treatment and control at week 4 is largely due to an increase in treatment measurements. (In my fake data, there is an 'interaction' between Procedure and Time effects.)

Instead of using a sequence of paired tests, it might be easier to explore such additional comparisons as ad hoc tests using contrasts based on results from the kind of ANOVA I mentioned in my Comment. Furthermore, if variances are essentially the same between weeks 2 and 4, and between treatment and control sides, you might have higher power to discover additional differences using ANOVA results. That is because ad hoc comparisons use information about variances from all the data, not just from one or two columns at a time, so standard errors may be smaller.

Note: I simulated my fake data using the R code below.

set.seed(930)
sbj = 1:20;  e = rnorm(20, 0, 4)
t2 = round(rnorm(20, 16, 2)+e, 2); t4 = round(rnorm(20, 18, 2)+e, 2)
c2 = round(rnorm(20, 12, 2)+e, 2); c4 = round(rnorm(20, 12, 2)+e, 2)
cbind(sbj, c2, c4, t2, t4)
colMeans(cbind(sbj, c2, c4, t2, t4)[,2:5])
c2      c4      t2      t4
11.9890 12.4065 15.4440 17.5190

• Thank you very much, Professor. Your explanation and example made it more clearer to me. But if my outcome data is not normally distributed, can I use ANOVA? Oct 2 '18 at 10:54
• You asked about t tests vs. ANOVA, so I supposed your data to be normal. If data are far from normal, you should use nonparametric alternatives to paired t test, perhaps Wilcoxon signed rank tests. // Maybe start new question showing some data, and ask what analysis to use. Oct 2 '18 at 16:45
• I'm really sorry to make you misunderstood. I would like to ask that if, at first, I disigned my RCT study and planned to use repeated ANOVA to analyse the data, but when the data was completely collected, it turned out that the data was not normally distributed. What should I do? Can I still use repeated ANOVA or do I have choose another tests? Is there any non-parametric test for repeated ANOVA? Thank you very much Professor Oct 3 '18 at 9:22