# Survey Data in General linear Models

I have survey data and a collection of covariates.

I have a few questions.

I am wanting to predict a dependent continuous variable with GLM (Ancova)

Is it necessary for this dependent variable to be normally distributed or does this assumption only apply to the residuals?

Additionally, survey data is usually interval in nature. For example, the score ranges from 0 to 60 on this particular task (ex. scores: 5, 16, 60, etc.). Can this be used in the model? Given the nature of the scale, most scores are on the lower side (frequency distribution of scores appears left-skewed) as this survery was given to a healthy population and no one has the psychopathology on the right extreme.

I am wanting to predict objectively measured food intake (kcal) while controlling for a priori determinants such as Fat free mass.

My model will look like as follows: Food Intake = Age + Sex + Fat Free Mass + IDS-R score

Where IDS-R is the inventory for depressive symtpology. I predict that a higher IDS-R score is associated with a higher food intake controlling for age, sex, and fat free mass. For this reason, I expect the parameter estimate (beta coefficient) to be positive.

Lastly, what happens if I see an interaction between Sex and IDS-R? For example, Sex was coded as Male = 1 and Female =2. The Sex beta coefficient is -200kcal; I interpreted this as men had lower food intake than women. However, the IDS-R * Sex interaction is a positive beta coefficient - how does get interpreted?

If anyone can assess my understanding of statistics and gaps in knowledge, can you please direct me to a useful resource regarding these applied concepts?

• user158565 Is it correct to use tags like multiple regression for the GlM mode? – Subhash C. Davar Jul 28 at 16:08

In a multiple linear regression model such as the one you listed, usually one checks whether the model residuals are normally distributed. This provides an indirect way of checking whether the distribution of Food Intake for subjects with specific values of Age, Sex, Fat Free Mass and IDS-R Score is normal.

See this post for a nice discussion on when you should worry about the dependent variable having theoretical bounds (in your case, 0 and 60): https://www.theanalysisfactor.com/linear-regression-outcome-boundaries/. However, note that the post contains an incorrect assertion: "it turns out, the distributions of the predictors don’t generally matter. It’s the outcome variable whose distribution matters." In reality, what matters is the distribution of the outcome (or dependent) variable corresponding to given values of the predictor variables - there's an expectation that that conditional distribution would be normal.

The interpretation of an interaction depends on how the Sex variable is coded in your data and how your statistical software treats that coding by default. To make things simpler, I will assume that your model contains a single interaction (i.e., Sex by IDS-R) and that Sex is coded in the data as 0 = Male and 1 = Female. I will also assume that your software's default when it comes to such coding (referred to as dummy coding) is to treat 0 as the reference category of Sex against which the remaining category of Sex (i.e., the one coded as 1) will be compared. Under these assumptions, your fitted regression model can be stated as:

Expected Food Intake = b0 + b1*Age + b2*Sex + b3*Fat Free Mass + b4*IDS-R + b5*Sex*IDS-R (1)

where the star symbol in equation (1) refers to multiplication and the word "expected" could be replaced with "mean" or "average". The regression coefficients b0 to b5 will be reported by your software (possibly in a different order than what I have here).

Since Sex is coded as either 0 or 1, we can use equation (1) to express the expected food intake for males (for whom Sex = 0) and females (for whom Sex = 1), respectively. Specifically, plugging 0 and 1 instead of Sex in turns in equation (1) gives the following:

Expected Food Intake for Males = b0 + b1*Age + b3*Fat Free Mass + b4*IDS-R (2)

Expected Food Intake for Females = (b0 + b2) + b1*Age + b3*Fat Free Mass + (b4 + b5)*IDS-R (3)

From these sub-models, we can see that:

• b4 represents the estimated effect of IDS-R on expected food intake for males having the same age and the same fat free mass;
• b4 + b5 represents the estimated effect of IDS-R on expected food intake for females having the same age and the same fat free mass.

Thus, if we return to equation (1), we can conclude that the coefficient of IDS-R (namely, b4) in that equation represents the effect of IDS-R on expected food intake for males having the same age and the same fat free mass. In other words, among males having the same age and the same fat free mass, each 1-point increase in IDS-R score is associated with b4 units change in expected food intake. (If b4 is positive, the change will be an increase; if b4 is negative, the change will be a decrease). In this sense, b4 is the rate of change in the expected food intake value associated with a 1-unit increase in IDS-R for males having the same age and the same fat free mass.

The coefficient b5 of Sex*IDS-R in equation (1) is the difference between b4 + b5 and b4: b5 = (b4 + b5) - b4. The meaning of b4 + b5 comes from equation (3) and that of b4 comes from equation (2):

• b4 + b5 is the rate of change in the expected food intake value associated with a 1-unit change in IDS-R for females having the same age and the same fat free mass;
• b4 is the rate of change in the expected food intake value associated with a 1-unit change in IDS-R for males having the same age and the same fat free mass.

The difference between b4 + b5 and b4 (that is, b5) is simply a difference between females and males in their rate of change in expected food intake value per 1-unit increase in IDS-R, assuming they have the same age and the same fat free mass.

If b5 is strictly positive, then b4 + b5 > b4, suggesting that the expected food intake changes (linearly) at a faster rate for females than males as IDS-R increases (all else being equal).

If b5 is strictly negative, then b4 + b5 < b4, suggesting that the expected food intake changes (linearly) at a slower rate for females than males as IDS-R increases (all else being equal).

Anyway, I probably over-complicated this explanation, but it hopefully gives you enough information for you to understand how you can interpret your regression coefficients.

Let's assume your model contains an interaction between Age and IDS-R (instead of Gender and IDS-R), where Age is treated as a continuous variable. Then the fitted model equation would be written as:

Expected Food Intake = c0 + c1*Age + c2*Sex + c3*Fat Free Mass + c4*IDS-R + c5*Age*IDS-R (4)

where the regression coefficients are denoted by c0 through c5 to make it clear they are going to be different from the regression coefficients b0 through b5 used earlier (hence they will have different values and different interpretations).

Equation (4) can be re-written in a way which will make it clear how the effect of IDS-R on expected food intake depends on Age, all else being equal. Specifically, grouping all terms which include IDS-R together, we obtain:

Expected Food Intake = c0 + c1*Age + c2*Sex + c3*Fat Free Mass + (c4 + c5*Age)*IDS-R (5)

The coefficient of IDS-R in equation (5) - namely, c4 + c5*Age - represents the rate of change in expected food intake per 1-unit increase in IDS-R score corresponding to subjects with the same sex and same fat free mass.

To meaningfully describe this rate of change, you can pick a few representative values of Age and compute the coefficient c4 + c5*Age for those values. (Recall that c4 and c5 are estimated from the data and reported in the model summary output, hence known.) To this end, plot the distribution of observed Age values in your data and pick some values of Age that are meaningful in connection to that distribution. For example, if the distribution is approximately normal, you can pick these 3 values:

• Age = Mean - SD
• Age = Mean
• Age = Mean + SD

where Mean stands for the average age in your data and SD stands for the standard deviation of the ages in your data. For example, Mean could be 40 (years) and SD could be 5 (years).

Then you would say:

For subjects with the same sex and same fat free mass, the rate of change in expected food intake per 1-unit increase in IDS-R score depends (linearly) on Age. In particular, among those subjects with an age of 40 - 5 = 35, expected food intake changes by c4 + c5*35 units for each extra IDS-R point. (The change will be an increase if the calculated value of c4 + c5*35 is strictly positive and a decrease if that value is strictly negative. Just plug in the actual values of c4 and c5 into c4 + c5*35 to get the rate of change. Same for the subsequent rates of change.) Among those subjects with an age of 40, expected food intake changes by c4 + c5*40 units for each extra IDS-R point. Among those subjects with an age of 40 + 5 = 45, expected food intake changes by c4 + c5*45 units for each extra IDS-R point.

• Thank you for your detailed explanation. Am I correct to assume the interaction term is interpreted in the same way if it is the product of two continuous covariates . – mindhabits Jul 28 at 0:19
• You're welcome, @mindhabits! The Addendum I added to my initial answer explains how you would interpret a two-way interaction between two continuous predictor variables. – Isabella Ghement Jul 28 at 2:11
• See www3.nd.edu/~rwilliam/stats2/l55.pdf for more details on how to interpret an interaction between two continuous predictors. – Isabella Ghement Jul 28 at 3:14