"However, considering the coefficient and its corresponding p-value of the group variable, I am unsure whether it would be appropriate to conclude that there is no significant difference between the groups." No, you do not know that for sure yet. The difference 20.95 holds for the covariate being 0, but if the covariate is 1, the difference would be 20.95 + 14.14 = 35.09. In order to see if this difference of 35.09 would be significant, you could subtract 1 from the covariate (newcov <- covariate - 1) and run the model again with newcov as covariate. This works fine, since the effect of groupTreatment is the effect for newcov=0, meaning covariate=1.
Suppose the effect of groupTreatment would be significant in that new model, then there must be a particular value of the covariate, somewhere between 0 and 1, for which the group effect gets significant. Suppose you would like to know if that value is 0.6, then you could subtract value 0.6 from your original covariate (newcov <- covariate - 0.6) and run the model again with newcov as covariate. The idea is: the effect of "groupTreatment" shown in the summary is the effect for value 0 of "newcov", whichever original value of "covariate" that may be. There may be an R package to find the precise value for the "turning point" of the covariate, i.e. the value for which the group effect gets significant. Below is an R script which show generates data and shows the estimates of the group (g) effect, the covariate (x2) effect and the interaction effect (g:x2) for all values of 0 through 9 of the covariate. Within the for loop, each consecutive value from 0 through 9 is subtracted from the original covariate x, to obtain x2:
set.seed(245)
g <- c(rep(0,100), rep(1,100))
x <- ifelse(g==0, runif(100,-0.5,7.5), runif(100,2.5,9.5))
x <- round(x)
table(x)
y <- 0.1*g + 0.2*x + 0.2*g*x + rnorm(10,0,3)
for (i in 0:9){
print(i)
x2 <- x - i
model <- lm(y ~ g*x2)
print(summary(model))
}
[1] 0
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.2471 0.4336 -2.876 0.00447 **
g 0.4980 0.8590 0.580 0.56275
x2 0.2570 0.1072 2.397 0.01747 *
g:x2 0.1084 0.1587 0.683 0.49523
---
[1] 1
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9901 0.3499 -2.829 0.00515 **
g 0.6064 0.7225 0.839 0.40234
x2 0.2570 0.1072 2.397 0.01747 *
g:x2 0.1084 0.1587 0.683 0.49523
---
[1] 2
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.7330 0.2827 -2.593 0.0102 *
g 0.7148 0.5972 1.197 0.2328
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 3
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.4760 0.2457 -1.937 0.0541 .
g 0.8233 0.4916 1.675 0.0956 .
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 4
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.2190 0.2526 -0.867 0.3871
g 0.9317 0.4207 2.215 0.0279 *
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 5
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03806 0.30044 0.127 0.8993
g 1.04010 0.40328 2.579 0.0106 *
x2 0.25703 0.10723 2.397 0.0175 *
g:x2 0.10842 0.15868 0.683 0.4952
---
[1] 6
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.2951 0.3738 0.789 0.4308
g 1.1485 0.4457 2.577 0.0107 *
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 7
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.5521 0.4606 1.199 0.2321
g 1.2570 0.5339 2.354 0.0196 *
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 8
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.8091 0.5546 1.459 0.1462
g 1.3654 0.6495 2.102 0.0368 *
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
[1] 9
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.0662 0.6527 1.633 0.1040
g 1.4738 0.7803 1.889 0.0604 .
x2 0.2570 0.1072 2.397 0.0175 *
g:x2 0.1084 0.1587 0.683 0.4952
---
The significance of the group (g) effect decreases to values below 0.05 and then increases again. So, for some values of the covariate there is a significant difference between the two groups of g (alpha=0.05). (The decrease and increase of the p-value of the group (g) effect over the range of values of covariate x2 is caused by the fact that the squared std. error of the group (g) effect is a quadratic function of the covariate value).
A final remark about your last question: "If I cannot draw such a conclusion based on this model, which model or which number should I examine to determine if there is a significant difference between the control and treatment groups?". If your data are OK as far as the assumptions for regression are met, it's best to stop at this point and not look for further analysis techniques. Your question sounds like "I definitely want to show that my treatment has a significant effect no matter what statistical method I have to use". Sometimes, an effect simply does not exist contrary to your expectations!
