Question

3. The iodine uptake by the coturnix quail thyroid gland has been examined by researchers at Ohio State University. In the course of the study, three measurements were taken at different time points Xi 24 hrs, X2 48 hrs, X3= 96 hrs at different levels of two factors: Thyroid states (hypothyroid, euthyroid, and hyperthyroid) and Temperature (20 C and 35° C). The measurements were repeated 6 times at each combination of the factor levels MANOVA table Sources of variation df (a) Identify the sources of variation and complete the corresponding degrees of freedom column for the above experiment. [6 pts.] (b) Write down the complete model for the above multivariate analysis of variance. [6 pts.]

Answer 1

ANSWER:

          The sources of variation can be -

             error (within-group) variability

          Variability due to Thyroid states group

           Variability due to different temperatures

Variability due to communication amid infection and thyroid state

       The degrees of freedom (df) -

          n = number of observations

           m = number of groups

null hypothesis - There is no important alteration in means of dissimilar groups

           dfT = n – 1, dfH = m – 1, dfE = n – m. As usual dfT = dfH + dfE

• The proposition mean squares (H) is comparable to between group mean squares of ANOVA

               And the error mean squares (E) which is equivalent to within group mean squares of ANOVA

                               Total df = Error df + hypothesis df

The model for above MANOVA -

                  Xikr = µ + αi + τ k + γik + eikr

Factor 1 (Thyroid states) has 3 levels

factor 2 (Temperatures) has 2 levels

The number of dependent variables is 3

• Xikr is the 3 × 1 vector of measurements on the rth unit in the ith level of factor 1 and the kth level of factor 2: , with i = 1, 2 , 3, k = 1, 2 and r = 1, 2, 3 and all are 3 × 1 vectors.
• Here γik is a vector of interaction effects. To simplify notation, for 3 units in each of the 6 combinations of factor levels, we have 18 units in each factor combination