# Difference or proportional

A report was claiming that older individuals were not appreciated at technology firms.

They studied a dozen factory locations of a tech firm where employees could (anonymously) suggest process improvements on a bulletin board in the hallway. Others could walk by and sign their name if they agreed.

The researchers knew which employee suggested what so they could study the effect of age. A dichotomous measure that is unity if a suggestion was written by an employee over the age of 35.

They find that suggestions by older employees received 8% fewer endorsements.

My concern is that they also state that 80% of the employees are under 35, and a similar proportion (71%) of the suggested improvements are from young employees. So, don't they have to show that the number of endorsements is fewer than what we would expect given the fact that the majority of employees are young? I feel like the result is being driven by the makeup of the firm, the board on any given day is dominated by young employee suggestions. On the other hand, it is not like employees are picking their favorite and then we could expect the proportion to matter.

I am not sure so I would like to know if it matters and if not why doesn't it? If it does how would they fix it? And, if it does is there an example of a study that does it right?

• You'll need to provide a link to the study. – Sycorax May 4 '15 at 18:12
• @user777 Can't find it unfortunately. The question stands regardless. Given this set up those are my questions. Treat it as a thought experiment – CJ12 May 4 '15 at 18:21

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.

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.

set.seed(1969)
n   <- 2000
beta    <- c(0.5, .15)
X   <- cbind(rep(1, n), rbinom(n, size=1, prob=0.2))
lambda  <- exp(X%*%beta)
y   <- rpois(n, lambda=lambda)
>    Estimate Std. Error z value Pr(>|z|)
X1  0.49279    0.01954  25.219  < 2e-16 ***
X2  0.17504    0.04079   4.291 1.78e-05 ***


As we can see, the model estimates the effect of age at roughly $0.17$, which is within 1 standard error of its true value, $0.15$.

But this doesn't necessarily mean that the study was done correctly. In my example, we precisely know the form of the data generating process, so it's very easy for us to re-create it. If there are other features influencing endorsements, then the authors' model suffers from omitted variable bias, and the coefficient estimates will be biased and inconsistent.

I suspect this regression, or some related variant, is what the researchers used. But this doesn't necessarily mean that the effect is as large as they stated (if it exists at all!). Perhaps the researchers selected the $35$ year age cut-off after examining the data, or any other kind of bias in the model construction step created the result ("The Garden of Forking Paths").

More subtly, it's possible that the trend is not binary at all but rather varies continuously and non-monotonically over one's life and is interactive with other worker characteristics. For example, it's possible that, as a function of age, there is some complex relationship as below. In this case, the estimates from the binary model will still be perfectly valid, but will not characterize well the kind of variation actually exists. A more flexible model, such as with splines, would be able to capture this.

• This makes sense. I thought the skew, or underlying proportion would matter. They use an normal regression because as I remember the number of endorsements is very skewed. So would they need to use a GLM? Is that the crux of them being right. Also, I forget what, but they control for about one dozen variables so there is less of a chance of leaving something out – CJ12 May 4 '15 at 19:14
• Also can you provide a cite for the claim about the GLM model not having that assumption. If so, a comment on the above and the cite and I will accept – CJ12 May 4 '15 at 19:16
• Great. One final question. If a continuous age variable was used, is the assumption the same as if it was binary. Your graph made me think of this – CJ12 May 6 '15 at 2:34
• Didnt know that, I did mean to award it. I just meant in the above that in your answer you said these models do not assume a distribution of the indicator age variable. Is this also true of age was continuous and not broken into one variable? Also, though it does not assume a distribution couldnt a skewed independent variable still affect the results? If so, is there a fix for this? – CJ12 May 6 '15 at 17:02
• Regression does not make assumptions about the distribution of the independent variables. But if there is some correlation between the IV and the DV, then the distribution of the IV will be reflected indirectly in the marginal distribution of the DC. But this doesn't matter for regression because regression cares about the conditional distribution of the DV, given the observed values of the IV(s). – Sycorax May 6 '15 at 17:05