Question

Some researchers tested whether arthritis in dogs could be improved by supplementation with antioxidants and/or an aminosugar mixture (containing glucosamine and chondroitin). They gave combinations of these supplements (each a factor with two levels: treatment and control) to equal numbers of test subjects in a balanced factorial design. They tested the effects of these supplements on levels of inflammation using a factorial ANOVA. The ANOVA table from their output is copied below.                                                                               

> model<-aov(inflammation~as.factor(antioxidant)*as.factor(aminosugar))

> anova(model)

	Df	SS	MS	F	P
antioxidant	1	385	385	17.5	0.0007
aminosugar	1	0.7	0.7	0.032	0.8581
antioxidant:aminosugar	1	1.3	1.3	0.059	0.7863
Residual	16	352	22

a) How many hypotheses did they test with this model?   

b) How many test subjects (i.e. replicates) did the researchers have?

c) Did the order in which antioxidant and aminosugar effects entered this model affect their significance? Why?   

d) Name a measure of model fit that can be used to compare the relative fit of different models, while taking into account the number of parameters in each?

e) The researchers simplified their model by removing the interaction term and the main effect of aminosugar. Fill in the table below with the values of their new model.   

> model2<-aov(inflammation~as.factor(antioxidant))

> ANOVA(model)

	Df	SS	MS	F
antioxidant
Residual

f) Did the significance of antioxidant change by removing the other terms, and if so, did it become more or less significant?

g) By how much did the overall model R² change (explain whether it increased, decreased or no change, as well as the amount)?    (show your working)   

Answer 1

a. This is a case of Two way ANOVA

Here, our response variable is Inflammation

our predictor variables are : Antioxidant ( levels =2)

Aminosugar ( levels = 2)

We also assume that there is interaction effect present between our independent predictor variables.

Hence, Our 3 hypotheses for this problem:

H01: Both the antioxidant groups have equal effect in improving arthritis by supplementation v/s   

H11: Both the antioxidant groups do not have equal effect in improving arthritis by supplementation

H02: Both the Aminosugar groups have equal effect in improving arthritis by supplementation v/s

H12 : Both the Aminosugar groups do not have equal effect un improving arthritis by supplementation

H03: Aminosugar and Antioxidant are independent or that interaction effect is not present v/s

H13: Aminosugar and Antioxidant are dependent or that interaction effect is present.

b. No of test subject the researcher have :

= Degree of freedom of Antioxidant + Degree of freedom of Aminosugar + degree of freedom for interaction + degrees of freedom of residual +1

= 1 + 1 +1+16+1 = 20

c. No, the order in which aminosugar and antioxidant effects are entered do not cause any affect in the level of significance . The reason for this is that the factors are nominal in nature and hence they do not possess any natural ordering . Hence there is no change in significance due to the order in which the factors has been entered.

d.Our measure of model fit for different models to compare relative fit taking in consideration the no. of parameters is

Adjusted R²

where k is the number of parameters. Now, R2 adjusted is maximum when optimal no. of predictors are used to explain the response variable. More the value of R^2_adj , better will be the model. Hence we keep on adding or deleting the no. of predictors in order to get the correct no. of predictors for our model, so that we get maximum adjusted R² with optimal no. of predictors. for our regression.