# Three continous variables + 2 factors vs. five continous variables to control for confounders?

I am trying to make sense of the design for my Master's thesis.

I am looking at how three different types of play relate to anxiety in children. So I have three continuous independent variables measured in hours spent on each type, and I also have two potential confounders potentially having an effect on anxiety that I want to account for which are also continuous (scores on a scale).

What would be the best design to account for the individual effects of my three independent variables of interest while at the same time levelling out the effects of the two confounders? Would it be helpful to run a multiple regression where it would show the individual contribution of each of the five independent variables? Alternatively, I could perhaps stratify the two confounding variables and assign all participants to different levels across the two factors, so then my design will have two factors and three continuous variables to look at, which seems overcomplicated. Please advise on the potential approach here. Thanks.

• A few questions to help you give us more background information. What is the dependent variable and how is it coded? If it doesn't have a smooth semi-symmetric distribution how did you plan to model it? What is the probability that confounding has been captured by the variables you have available? What is the sample size? What is the entire frequency distribution of the outcome variable? Commented Apr 6 at 11:33
• Thanks for the reply. The dependent variable is the level of anxiety measured by RCADS scale where participants rate items on a 4-point Likert scale. The final raw score is then obtained by adding up all the points and ranges between 0-30. The sample size is supposed to be > 460. I’m not sure about the data distribution questions and how to measure it at this stage. As for the confounders, those will be parental depression and anxiety score, stressful events score and a child’s age that I believe could impact the outcome variable alongside the three IVs and need to be adjusted for. Commented Apr 6 at 14:45

The effective sample size at your disposal for estimating model parameters will depend on the actual distribution of the 0-30 dependent variable, especially where there are a lot of ties creating a floor or ceiling effect. The best bet for analysis, if the sample size is sufficient, is an ordinal semiparametric regression model such as the proportional odds model. Resources for studying this model are here. The R rms package lrm and orm functions are two functions set up for this, and you can also look at the R ordinal and VGAM packages.

• Thank you very much for the detailed guidelines! Unfortunately, the link is taking to a non-existent page :( Do I understand correctly that the semiparametric regression model is recommended because the dependent variable is based on a Likert-type scale, which can't be treated as metric data? Or would it still be possible to use a linear regression if the distribution approached normality? Commented Apr 9 at 12:18
• Sorry; link is now fixed. You won't find normality with constant variance for discrete Likert scales so that's not a very fruitful pursuit. Commented Apr 9 at 12:44
• I see now. Thank you very much! Commented Apr 9 at 13:13

Frank Harrell's suggestion for ordinal regression here makes a lot of sense. (+1)

In general it's not a good idea to bin continuous predictor variables; see this page and its links. Multiple regression including all continuous predictors would be the best choice.

As Professor Harrell explains in Multiple Regression Strategies, fitting regression splines for the continuous predictors will allow you to avoid both binning the predictors and imposing an unrealistic linear relationship between the predictors and outcome (in the scale used for ordinal regression). With 460+ cases that should be possible for all 5 predictors without overfitting, although with ordinal outcomes the details of the outcome distribution can affect the necessary sample size.

• Thanks very much for the response. So do we treat data obtained from Likert-type scales as ordinal? I am also a bit confused with the point where you're saying that with ordinal outcomes the details of the outcome distribution can affect the necessary sample size. Could you please explain that? I calculated my needed sample size based on power analysis and was going to check the distribution after data has been collected, how is it possible to adjust sample size based on the distribution? Commented Apr 9 at 12:53
• @Ksenia the best way to treat Likert-type outcome data like yours is ordinal. I suspect that your power calculation was based on a continuous outcome and ordinary least squares, which might not be correct for such data. Section 4.4 of the Harrell reference shows a formula for the number of observations needed just to avoid overfitting with an ordinal outcome. It depends on the number of cases at each outcome level. What you might need to detect a specific magnitude of effect could well be more.
– EdM
Commented Apr 9 at 14:03
• And for sample size calculations done explicitly for ordinal variables see this. Commented Apr 10 at 13:09
• Thanks very much!! Commented Apr 11 at 13:35