Question

1. You are interested in whether people who exercise while tired feel better about their bodies than people who exercise while not tired. You also measure the body image evaluations of non-exercises (separately for tired vs. not tired participants). You do not think that non-exercisers differ in the perceived positivity of their body image depending on whether they are tired or not. You also think that exercisers may have better body images than non-exercisers.
   1. What kind of design would you use to analyze your data for the above example? (Use the correct notation). 4 points
   2. Draw the predictions outlined above, using a bar graph. Label all variables and levels. (8 points)
   3. Use the names of the variables from this example to explain what a main effect is. What are your predictions for each of the main effects? (4 points)
   4. Use the names of the variables from this example to explain what an interaction is. What is your prediction for the interaction? (4 points)

Answer 1

solution:

According to the question there are two factors here namely Exercisers and Non-Exercisers and both of them have two levels namely tired and not tired therefore this is a 2*2 factorial design

It looks almost the same as the randomized block design model only now we are including an interaction term:

i=1.........a

j=1..........b

k=1..........n

Thus we have two factors in a factorial structure with n observations per cell. As usual, we assume the , i.e. independently and identically distributed with the normal distribution.

The cell means model is written:

Here the cell means are: . Therefore we have a × b cell means, μij.

the marginal means are defined as the simple average of those means

It follows that the interaction terms are defined as the difference between our cell means and the additive portion of the model:

There is really no benefit to the effects model when there is interaction, except that it gives us a mechanism for partitioning the variation due to the two treatments and their interactions. Both models have the same number of distinct parameters.

Now, we'll take a look at the strategy for deciding whether our model fits, whether the assumptions are satisfied and then decide whether we can go forward with an interaction model or an additive model. This is the first decision. When you can eliminate the interactions because they are not significantly different from zero, then you can use the simpler additive model. This should be the goal whenever possible because then you have fewer parameters to estimate, and a simpler structure to represent the underlying scientific process.

hence in order to test significance this is the following hypothesis

H0: vs H1:at least one of those effects is not equal to zero.

If the interaction term is significant that tells us that the effect of A is different at each level of B. Or you can say it the other way, the effect of B differs at each level of A. Therefore, when we have significant interaction, it is not very sensible to even be talking about the main effect of A and B, because these change depending on the level of the other factor. If the interaction is significant then we want to estimate and focus our attention on the cell means. If the interaction is not significant, then we can test the main effects and focus on the main effect means.

the main effects and the predictions for them as well as the interactions and the predictions are included in the analysis above