One-way ANOVA

Mice experiment: lifetime versus diet

The data for this discussion comes from the book The Statistical Sleuth, by Ramsey and Schafer (Duxbury, 1997). An experiment was conducted on female mice, which were randomly assigned to one of the following six treatment groups (as described by Ramsey and Schafer):

  1. NP: Mice in this group ate as much as they pleased of a nonpurified, standard diet for laboratory mice.
  2. N/N85: This group was fed normally both before and after weaning. (The slash distinguishes the two periods.) After weaning, the ration was controlled at 85 kcal/wk. This, rather than NP, serves as the control group because caloric intake is held reasonably constant.
  3. N/R50: This group was fed a normal diet before weaning and a reduced-calorie diet of 50 kcal/wk after weaning.
  4. R/R50: This group was fed a reduced-calorie diet of 50 kcal/wk both before and after weaning.
  5. N/R50 lopro: This group was fed a normal diet before weaning, a restricted diet of 50 kcal/wk after weaning, and had dietary protein content decreased with advancing age.
  6. N/R40: This group was fed normally before weaning and was given a severely reduced diet of 40 kcal/wk after weaning.

NOTE: This data comes from R. Weindruch, R.L. Walford, S. Fligiel, and D. Guthrie, "The Retardation of Aging in Mice by Dietary Restriction: Longevity, Cancer, Immunity and Lifetime Energy Intake," Journal of Nutrition 116(4) (1986): 641-54.

Generally the researchers' questions concerned comparisons of lifetimes between the various diets. There were specifically these questions:

  1. Does lifetime for the 50 kcal/wk diet exceed that for the 85 kcal/wk diet, and, if so, by how much, (This calls for a comparison of the N/R50 group to the N/N85group.)
  2. Is lifetime affected by providing a reduced calorie diet before weaning, given that a 50 kcal/wk diet is provided after weaning? (This calls for a comparison of the R/R50 group to the N/R50 group.)
  3. Does lifetime on the 40 kcal/wk diet exceed the lifetime on the 50 kcal/wk diet? (This calls for a comparison of the N/R40 group to the N/R50 group.)
  4. Given a reduced calorie intake diet of 50 kcal/wk, is there any additional effect on lifetime due to decreasing the protein intake? (This calls for a comparison of the N/R lopro diet to the N/R50 diet.)
  5. Is there an effect on lifetime due to restriction at 85 kcal/week? This would indicate the extent to which the 85 kcal/wk diet served as a proper control and possibly whether there was any effect of restricting the diet, even with a standard caloric intake. (This calls for a comparison of the N/N85 group to the NP group.)


These data are available as the data frame mice.df which should be available to you automatically from my account. The data frame contains two variables: Lifetime, which gives the lifetime of each mouse in months, and Treatment, whose values are described above.


One-way ANOVA with R

Get the data and do basic descriptive statistics:
attach(mice.df)
summary(mice.df)
boxplot(Lifetime ~ Treatment)
nn <- tapply(Lifetime, Treatment, length)
mm <- tapply(Lifetime, Treatment, mean)
ss <- tapply(Lifetime, Treatment, sd)
round(cbind(nn,mm,ss),1)

Do the one-way ANOVA:
fm1 <-aov(Lifetime ~ Treatment) #Fit the one-way ANOVA model.
anova(fm1) #Obtain the ANOVA table.

Follow-up analysis:
pairwise.t.test(Lifetime, Treatment, p.adj="bonferroni")

Here is the Tukey analysis for the mice.short.df data. 


                diff    lcl    ucl  conclusion
NN85-NP         4.90   3.29   6.51    reject
NN85-NR40     -13.94 -15.55 -12.33    reject
NN85-NR50      -9.00 -10.61  -7.39    reject
NN85-LOPRO    -10.66 -12.27  -9.05    reject
NN85-RR50      -8.66 -10.27  -7.05    reject
NP-NR40       -18.84 -20.45 -17.23    reject
NP-NR50       -13.90 -15.51 -12.29    reject
NP-LOPRO      -15.56 -17.17 -13.95    reject
NP-RR50       -13.56 -15.17 -11.95    reject
NR40-NR50       4.94   3.33   6.55    reject
NR40-LOPRO      3.28   1.67   4.89    reject
NR40-RR50       5.28   3.67   6.89    reject
NR50-LOPRO     -1.66  -3.27  -0.05    reject
NR50-RR50       0.34  -1.27   1.95    NS
LOPRO-RR50      2.00   0.39   3.61    reject
Here is the Tukey analysis for full mice data set. 
> round(cbind(nn,mm,ss),2)
            nn    mm   ss
N/N85       57 32.69 5.13
NP          49 27.40 6.13
N/R40       60 45.12 6.70
N/R50       71 41.91 7.58
N/R50_lopro 56 42.89 6.68
R/R50       56 40.17 7.40


mu_i  mu_j    Y-bar_i - Y-bar_j   Signif(.05)?
---------------------------------
NN85  NP	  5.29		    yes
NN85  NR40	-12.43		    yes	
NN85  NR50	 -9.22		    yes
NN85  LOPRO	-10.22		    yes
NN85  RR50	 -7.48		    yes
NP    NR40	-17.72		    yes
NP    NR50	-14.51		    yes
NP    LOPRO	-15.49		    yes
NP    RR50	-12.77		    yes
NR40  NR50	  3.21		    yes
NR40  LOPRO	  2.23		    no
NR40  RR50	  4.95		    yes
NR50  LOPRO	  -.98		    no
NR50  RR50	  1.74		    no
LOPRO RR50	  2.72		    no


             NP      NN85               RR50
                                          NR50
                                            LOPRO  NR40
              o         o                 o  o  o   o
             ___       ___                     _______
                                         _________
    ----|----------|----------|----------|----------|---
        25        30         35         40         45