Correct Stats to Run?

Question

I am doing a study on body image perceptions and physical activity among adolescents. My hypothesis is adolescents who engage regularly in physical activity will have a more accurate perception of body image in relation to less active people. I have collected data on gender, age, bmi, bmi categories, physical activity levels, body image perceptions. What are the best stats tests to run??

Answer 1

I wouldn't bother with the BMI categories. This sounds like a grouped continuous variable, and these tend to waste information and attenuate correlations ^{(see Anderson, 1984; Bollen & Barb, 1981)}. Your actual BMI data should provide richer information without tetrachotomization (i.e., splitting a continuum into four discrete groups).

Assuming your perception survey data are based on Likert ratings (which are ordinal data), you may want to consider a couple alternative approaches for dealing with ordinal data:

1. Calculate correlations between BMI and each survey item.

   • Kendall's $\tau$ seems like a good choice of estimate to recommend. See "How do the Goodman-Kruskal gamma and the Kendall tau or Spearman rho correlations compare?"

     • This won't require much data or be sensitive to nonnormality.
   • Pearson's r can work and may offer more powerful significance tests if you're looking to do one, but can be biased by odd distributions (see Anscombe's quartet).
   • This may produce some confusing results if items have unexpected idiosyncrasies that affect their relationships with BMI. This method does not control measurement error.

2. Analyze the latent structure of your perception survey data.

   • This requires at least three items to produce meaningful results with factor or principal components analysis or Very Simple Structure. At least four items would be necessary to estimate more than one latent factor. I'm not sure the same requirements apply to item response theory models.

   • If you can identify a clear latent factor structure (I'm guessing you'd want a simple solution such as a single latent factor that all items correlate with more or less equally and strongly), you can use it to guide measurement model specification. In the maximally simple scenario, you'd be able to treat each item equally and add or average them to produce a single factor score. It is more likely that some items measure the latent factor(s) better than others, and can be weighted accordingly in factor scoring. In any case, you can estimate the correlation(s) between BMI and your factor score(s) instead of items to produce an estimate of the overall relationship that controls measurement error somewhat.

     • Reliable estimates of factor loadings in your measurement model will also require a lot of observations of each variable. Since each additional item will necessitate estimating an additional set of covariances (unless you have a prior theory that can justify fixing some estimates to be equal to one another or a specific constant, e.g., zero), the number of observations you'll want will increase with the number of items you have. Sample size is less of a concern if you wish to apply classical test theory assumptions and treat all items as equally good or error-free indicators of your latent factor(s), but other concerns apply when adding these assumptions. For more on that, see my answer to "Correlational study or ordinal data using 5-point Likert scale".
3. Regress BMI onto multiple predictors to look for moderating or curvilinear relationships.
   • Some of your other variables might moderate relationships between BMI and perception items – gender seems a particularly likely candidate. Consider adding these variables and their interactions with BMI to models predicting perception.
   • BMI might have a curvilinear relationship with some perception items, depending on their nature. For example, I would expect a partially quadratic relationship between BMI and agreement with an item like, "I am happy with my weight." This can be assessed with multiple regression.
   • If some of your perception items can be compared directly to BMI to estimate perception-reality discrepancies (e.g., "I think I am obese" answered "Strongly agree" by someone with BMI < 20), these items might be worth including with BMI in models predicting responses to other items like the previous example. Interactions between perception items about physical facts and BMI could be particularly relevant for your work in general.
   • The most direct test of the hypothesis you mention would probably be predicting perception items or factor scores from BMI, physical activity level, and the interaction between these two variables. It seems you would want a stronger relationship between BMI and body image perception among people reporting more physical activity. If this is the case, your interaction term should be nonzero. Whether it's meaningfully different will depend on the strength of the relationship, the reliability of your data, and your standards for judging meaningful differences. How to handle your physical activity variable also depends on the level of measurement, which I neglected to ask about in my comment. I'll leave that matter unaddressed for now in case this really is a self-study question.

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^{References
Anderson, J. A. (1984). Regression and ordered categorical variables. Journal of the Royal Statistical Society B, 46, 1–30.
Bollen, K. A., & Barb, K. H. (1981). Pearson's $r$ and coarsely categorized measures. American Sociological Review, 46, 232–239. Retrieved from http://www.statpt.com/correlation/bollen_barb_1981.pdf.}