F test for multiple paired samples

Question

I have two sets of data: Set1 and Set2.

For each set we have the same variables A, B, C, D, E.

I want to do the F-test to understand whether the following relationships are simultaneously true:

Set1_A = Set2_A, Set1_B = Set2_B, Set1_C = Set2_C, Set1_D = Set2_D, Set1_E = Set2_E

Set1_A and Set2_A might be different size vectors.

How do I achieve this in R?

Thanks

Sample Data for Set1:

A       B       C
11.0    11.0    11.0
23.3    23.3    23.3
44.6    -1.3    -7.1
-1.9    -1.9    -1.9

Sample Data for Set2:

A        B      C
3.9      3.9    3.9
-6.1    -6.1    -6.1
-34.6   -95.7   -102.4
 7.0    7.0     7.0

Answer 1

This illustrates how to get a comparison for Set1_A vs Set2_A. In order to determine if they are simultaneous "true" you would need to use a multivariate analysis

Set1 <- read.table(text="A       B       C
 11.0    11.0    11.0
 23.3    23.3    23.3
 44.6    -1.3    -7.1
 -1.9    -1.9    -1.9", header=TRUE)

 Set2<- read.table(text="A        B      C
 3.9      3.9    3.9
 -6.1    -6.1    -6.1
 -34.6   -95.7   -102.4
  7.0    7.0     7.0", header=TRUE)
 combset <- rbind(Set1, Set2)
 combset$grp <- rep(c("Set1", "Set2"), times=c(nrow(Set1), nrow(Set2) ) )
 combset
#----------------
      A     B      C  grp
1  11.0  11.0   11.0 Set1
2  23.3  23.3   23.3 Set1
3  44.6  -1.3   -7.1 Set1
4  -1.9  -1.9   -1.9 Set1
5   3.9   3.9    3.9 Set2
6  -6.1  -6.1   -6.1 Set2
7 -34.6 -95.7 -102.4 Set2
8   7.0   7.0    7.0 Set2

Now that you have your data in what might be called the long format you can use the grp ID as a factor in an lm.formula call:

 lm(A ~ grp, data=combset)

Call:
lm(formula = A ~ grp, data = combset)

Coefficients:
(Intercept)      grpSet2  
      19.25       -26.70  

Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor
> anova(lm(A ~ grp, data=combset))
Analysis of Variance Table

Response: A
          Df Sum Sq Mean Sq F value  Pr(>F)  
grp        1 1425.8 1425.78  3.8004 0.09913 .
Residuals  6 2251.0  375.16                  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor

The multivariate model can be constructed. BUT ... are you sure you can interpret this correctly and are aware of the statistical issue that might arise?

>  lm( A + B + C ~ grp, combset)

Call:
lm(formula = A + B + C ~ grp, data = combset)

Coefficients:
(Intercept)      grpSet2  
      33.35       -87.92  

Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor
> anova(lm( A + B + C ~ grp, combset))
Analysis of Variance Table

Response: A + B + C
          Df Sum Sq Mean Sq F value Pr(>F)
grp        1  15462 15461.6   2.016 0.2055
Residuals  6  46017  7669.6               
Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor

I was worried about that answer because I thought that more coefficients should have been estimated. I remembered an article in RNews by Peter Dalgaard and looked it up. This should have been what I offered:

>  lm( cbind(A, B, C) ~ grp, combset) 

Call:
lm(formula = cbind(A, B, C) ~ grp, data = combset)

Coefficients:
             A        B        C      
(Intercept)   19.250    7.775    6.325
grpSet2      -26.700  -30.500  -30.725

Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor
> anova(lm( cbind(A, B, C) ~ grp, combset))
Analysis of Variance Table

            Df  Pillai approx F num Df den Df Pr(>F)
(Intercept)  1 0.51946  1.44130      3      4 0.3557
grp          1 0.42690  0.99318      3      4 0.4813
Residuals    6                                      
Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor
> class(lm( cbind(A, B, C) ~ grp, combset))
[1] "mlm" "lm" 
Warning message:
In model.matrix.default(mt, mf, contrasts) :
  variable 'grp' converted to a factor

Notice that "real" multivariate inferential statistics (e.g. Pillai's trace or Wilks or Hotelling) are presented and that three separate coefficients for A, B and C are presented, and that the class of the output is "mlm" and not just "lm". You should also look at ?anova.mlm.