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One way to study the problem would be some kind of GLM. For example, a Poisson or negative binomial regression with an intercept term and the $\{0,1\}$ variable for age is an obvious starting point. This model doesn't make any assumptions about how the age indicator variable is distributed, so it doesn't matter if it's heavily skewed in favor of younger or older workers. The coefficient estimate will tell you the direction and magnitude of age on the number of endorsements even if there's a disparity in workers' ages.

One way to study the problem would be some kind of GLM. For example, a Poisson or negative binomial regression with an intercept term and the $\{0,1\}$ variable for age is an obvious starting point. This model doesn't make any assumptions about how the age indicator variable is distributed, so it doesn't matter if it's heavily skewed in favor of younger or older workers. The coefficient estimate will tell you the direction and magnitude of age on the number of endorsements even if there's a disparity in workers' ages.

You write the that the researchers' response data were skewed. If the conditional expectation (the product $X\beta$ of the "true" model) of the response is non-normal (which skewness implies), then they are violating one of the key assumptions of OLS. A GLM may be a more appropriate model in that case, provided that the selected family and link function are correctly suited to the problem under study. Since the number of endorsements must be a non-negative positive integer, a normal model must be wrong because a normal model admits negative and non-integer values as valid numbers of endorsements.

But a definitive reference for regression assumptions can be found inf Gleman and Hill, Data Analysis Using Regression and Multilevel/Hierarchical Models, section 3.6. All assumptions are listed in decreasing order of importance. The distribution of the independent variables doesn't appear on the list.

You write the that the researchers' response data were skewed. If the conditional expectation (the product $X\beta$ of the "true" model) of the response is non-normal (which skewness implies), then they are violating one of the key assumptions of OLS. A GLM may be a more appropriate model in that case, provided that the selected family and link function are correctly suited to the problem under study. Since the number of endorsements must be a non-negative positive integer, a normal model must be wrong because a normal model admits negative and non-integer values as valid numbers of endorsements.

Here's a demonstration using some fake data generated according to the process you describe. (The code is for R.) The data generation process fixes our parameter of interest $0.15$, and the intercept at $0.5$; this is the vector beta. Then we generate a binary indicator for the age variable that is $1$ only about 80% of the time; including an intercept, we have X. Then we generate the outcome y (endorsements) conditional on the expected value lambda.

You write the that the researchers' response data were skewed. If the conditional expectation (the product $X\beta$ of the "true" model) of the response is non-normal (which skewness implies), then they are violating one of the key assumptions of OLS. A GLM may be a more appropriate model in that case, provided that the selected family and link function are correctly suited to the problem under study. Since the number of endorsements must be a non-negative positive integer, a normal model must be wrong because a normal model admits negative and non-integer values as valid numbers of endorsements.

Here's a demonstration using some fake data generated according to the process you describe. (The code is for R.) The data generation process fixes our parameter of interest $0.15$, and the intercept at $0.5$; this is the vector beta. Then we generate a binary indicator for the age variable that is $1$ only about 80% of the time; including an intercept, we have X. Then we generate the outcome y (endorsements) conditional on the expected value lambda.