Question

1. For each of these questions, your audience are persons that are not experts in statistics. Write with complete sentences and paragraphs. Cite any references that you use.

   1. (5 pts.) When building a model, you make four assumptions about the residuals. Explain what they are and how you can verify that your assumptions are correct.

   2. (5 pts) Define ‘interaction term’. From your own experience, identify an instance in which you believe an interaction term would be appropriate.

Answer 1

When building a model, you make four assumptions about the residuals. Explain what they are and how you can verify that your assumptions are correct.

There are four assumptions associated with a linear regression model:

1. Linearity: The relationship between X and the mean of Y is linear.
2. Homoscedasticity: The variance of residual is the same for any value of X.
3. Independence: Observations are independent of each other.
4. Normality: For any fixed value of X, Y is normally distributed.

Linearity – we draw a scatter plot of residuals and y values. Y values are taken on the vertical y-axis, and standardized residuals (SPSS calls them ZRESID) are then plotted on the horizontal x-axis. If the scatter plot follows a linear pattern (i.e. not a curvilinear pattern) that shows that linearity assumption is met.  

Independence – we worry about this when we have a longitudinal dataset. The longitudinal dataset is one where we collect observations from the same entity over time, for instance, stock price data – here we collect price info on the same stock i.e. same entity over time.

Normality: we draw a histogram of the residuals, and then examine the normality of the residuals. If the residuals are not skewed, that means that the assumption is satisfied.

Equality of variance: We look at the scatter plot which we drew for linearity (see above) – i.e. y on the vertical axis, and the ZRESID (standardized residuals) on the x-axis. If the residuals do not fan out in a triangular fashion that means that the equal variance assumption is met.

Define ‘interaction term’. From your own experience, identify an instance in which you believe an interaction term would be appropriate.

An interaction effect happens when one explanatory variable interacts with another explanatory variable on a response variable. This is opposed to the “main effect” which is the action of a single independent variable on the dependent variable.

For example, let’s say you were studying the effects of a diet drink and a diet pill (the explanatory variables) on weight loss. The “main effects” would be the effect of a diet drink on weight loss, and the effect of the diet pill on weight loss. The interaction effect happens when the drink and pill took at the same time. It’s possible the combination could speed up weight loss, or even slow it down. Synergy in medicine, where two drugs work together to produce an effect greater than their additive individual effects, is a special case of an interaction effect.