Question 1:
> library(ISwR)
> data(ashina)
> ashina$subject <- factor(1:16)
> act <- data.frame(vas = ashina$vas.active, subject = ashina$subject, treat = 1, period = ashina$grp)
> plac <- data.frame(vas = ashina$vas.plac, subject = ashina$subject, treat = 0, period = ashina$grp)
> combined_data <- rbind(act, plac)
> model <- lm(vas ~ subject + treat + period, data = combined_data)
> summary(model)
Call:
lm(formula = vas ~ subject + treat + period, data = combined_data)
Residuals:
Min 1Q Median 3Q Max
-48.94 -18.44 0.00 18.44 48.94
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) -113.06 27.39 -4.128 0.000895 ***
subject2 51.50 37.58 1.370 0.190721
subject3 121.50 37.58 3.233 0.005573 **
subject4 97.00 37.58 2.581 0.020867 *
subject5 125.00 37.58 3.326 0.004604 **
subject6 31.50 37.58 0.838 0.415070
subject7 119.50 37.58 3.180 0.006215 **
subject8 132.00 37.58 3.513 0.003142 **
subject9 80.50 37.58 2.142 0.049003 *
subject10 116.00 37.58 3.087 0.007518 **
subject11 121.50 37.58 3.233 0.005573 **
subject12 154.50 37.58 4.111 0.000925 ***
subject13 131.00 37.58 3.486 0.003318 **
subject14 125.00 37.58 3.326 0.004604 **
subject15 99.00 37.58 2.634 0.018768 *
subject16 80.50 37.58 2.142 0.049003 *
treat -42.87 13.29 -3.227 0.005644 **
period NA NA NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 37.58 on 15 degrees of freedom
Multiple R-squared: 0.7566, Adjusted R-squared: 0.4969
F-statistic: 2.914 on 16 and 15 DF, p-value: 0.02229
> t.test(ashina$vas.active, ashina$vas.plac, paired = TRUE)
Paired t-test
data: ashina$vas.active and ashina$vas.plac
t = -3.2269, df = 15, p-value = 0.005644
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-71.1946 -14.5554
sample estimates:
mean difference
-42.875
The treatment has a value of -42.87, stating that the active treatment reduces
pain scores. The people that go the active treatment 43% less pain than those who
got placebo. The p-value of treatment is 0.0056, which is less than 0.05, so it
is significant.
Each subject has an individual pain score, and there is variation as it goes from
31 all the way to 154. There was no impact on different periods.
The t-results' p-value is 0.005644 which is less than 0.05. It is significant
that the active treatment reduces pain compared to the placebo. The confidence
interval is from -71.19 to -14.56 so the pain reduction from the active
treatment is between 14.56 and 71.19
Question 2:
> a <- gl(2, 2, 8)
> b <- gl(2, 4, 8)
> x <- -1:8
> y <- c(1:4, 8:5)
> z <- rnorm(8)
> interaction_model_matrix <- model.matrix(z ~ a*b)
> interaction_model_matrix
(Intercept) a2 b2 a2:b2
1 1 0 0 0
2 1 0 0 0
3 1 1 0 0
4 1 1 0 0
5 1 0 1 0
6 1 0 1 0
7 1 1 1 1
8 1 1 1 1
attr(,"assign")
[1] 0 1 2 3
attr(,"contrasts")
attr(,"contrasts")$a
[1] "contr.treatment"
attr(,"contrasts")$b
[1] "contr.treatment"
> interaction_model <- lm (z ~ a*b)
> summary(interaction_model)
Call:
lm(formula = z ~ a * b)
Residuals:
1 2 3 4 5 6 7 8
-0.32576 0.32576 -0.03928 0.03928 -0.08263 0.08263 -0.11748 0.11748
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.6907 0.1791 3.857 0.018201 *
a2 -2.1479 0.2533 -8.481 0.001060 **
b2 -1.0594 0.2533 -4.183 0.013887 *
a2:b2 3.4278 0.3582 9.570 0.000666 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2533 on 4 degrees of freedom
Multiple R-squared: 0.9652, Adjusted R-squared: 0.939
F-statistic: 36.94 on 3 and 4 DF, p-value: 0.002249
> only_interaction_model_matrix <- model.matrix(~ a:b)
> only_interaction_model_matrix
(Intercept) a1:b1 a2:b1 a1:b2 a2:b2
1 1 1 0 0 0
2 1 1 0 0 0
3 1 0 1 0 0
4 1 0 1 0 0
5 1 0 0 1 0
6 1 0 0 1 0
7 1 0 0 0 1
8 1 0 0 0 1
attr(,"assign")
[1] 0 1 1 1 1
attr(,"contrasts")
attr(,"contrasts")$a
[1] "contr.treatment"
attr(,"contrasts")$b
[1] "contr.treatment"
> only_interaction_model <- lm (z ~ a:b)
> summary(only_interaction_model)
Call:
lm(formula = z ~ a:b)
Residuals:
1 2 3 4 5 6 7 8
-0.32576 0.32576 -0.03928 0.03928 -0.08263 0.08263 -0.11748 0.11748
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.9111 0.1791 5.088 0.007043 **
a1:b1 -0.2205 0.2533 -0.871 0.433121
a2:b1 -2.3684 0.2533 -9.351 0.000728 ***
a1:b2 -1.2799 0.2533 -5.053 0.007214 **
a2:b2 NA NA NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2533 on 4 degrees of freedom
Multiple R-squared: 0.9652, Adjusted R-squared: 0.939
F-statistic: 36.94 on 3 and 4 DF, p-value: 0.002249
The model with (z ~ a*b) includes the effects of a and b and the interaction
between them a:b. While the formula (z ~ a:b) only includes the interaction between
a and b, without the main effects of a or b. (z ~ a*b) is more flexible as it
includes both types of effects. (z ~ a:b) is more likely to involve singularities
because it depends on one interaction.
Comments
Post a Comment