5  Complicated regression

5.1 Dependencies in your data

5.1.1 What is independent data

Broadly, having knowledge about one observation should not give you knowledge about other observations in your dataset.

For example, if we flip a coin ten times, knowing the result of the first coin toss (heads) will not give us knowledge about the subsequent 9 coin tosses.

5.1.2 What is data with dependencies

In reality, most data have dependencies, so we will focus on some of the most important kinds that will severely impact your analyses if you do not identify them.

5.1.3 Repeated measures/longitudinal data

If you have a dataset with repeated measures (e.g. some people in your cohort have more than one observation), then the repeated observations on each person cause dependencies in the data. That is, if a person has their weight measured five times, then just by knowing their first weight you can have a good guess at what their subsequent weights will be.

5.1.4 Matched data

Inside case-control studies, for each case a control (or multiple cases) can be selected to have similar attributes. For example, for each case, a control can be selected with a similar age. These controls have been “matched” to a case, and have introduced dependencies into the data.

5.1.5 Grouped/clustered data

Repeated measures data is a type of clustered data, where each person is their own cluster. Matched data is also a type of clustered data, where each group of matched controls-to-a-case is a cluster.

There can be many other kinds of clusters, for example:

  • Data sampled from multiple hospitals could have the hospital as the cluster variable
  • Data sampled from multiple countries could have the country as the cluster variable
  • Data sampled from multiple counties/municipalities could have the country/municipality as the cluster variable
  • If it is a study of children, and multiple children from each mother are included, then the mother could be the cluster variable

5.2 Analysing data with dependencies

5.2.1 Mixed effects regression

Mixed effects regression is an extension of the simple regression models (fixed effects) that we learned about in the previous chapter. You can have:

  • Mixed effects linear regression
  • Mixed effects logistic regression
  • Mixed effects negative-binomial regression

Mixed effects models are models that have both fixed and random effects. “Effects” is the term used to describe the estimated impact a variable has on the outcome. For example, “the effect of smoking on the risk of lung cancer”.

To determine if an effect is fixed or random, we introduce the concept of pooling data (i.e. sharing strength). Let us assume we have sampled 100 people per city for 9 cities, and 2 people for 1 city, and we want to estimate the average height in each city, we can do one of the following three options:

  1. Within each city, take the average height of the 100 (or 2) sampled people (zero pooling)
  2. Take the average height of 1000 sampled people and say each city is the same height (complete pooling)
  3. Estimate how much the average height varies between the cities. If the average height doesn’t vary much, give estimates close to #2. If the average height varies a lot, give estimates close to #1 (partial pooling)

Zero pooling tends to work well when you have a limited amount of effects you want to estimate, each with a good sample size (i.e. you don’t need to borrow strength from other data points). Partial pooling tends to work well when you have a large number of effects you want to estimate, each with a small sample size (i.e. you need to borrow strength from other data points).

Fixed effects are estimated with zero pooling. Random effects are estimated with partial pooling.

With data that has a large number of clusters, the clustering need to be accounted for in the regression model. This is done by introducing a variable that uniquely identifies each cluster, and allows the cluster effect to be estimated (e.g. in Oslo people are 1% more likely to die than in the rest of Norway). If there are a large number of clusters with a small amount of people in each cluster, then the cluster variable should be estimated using partial pooling (i.e. random effects).

In summary: Mixed effects regression should be used for grouped/clustered/matched data (and subsequently repeated measures/longitudinal data).

5.2.2 Conditional logistic regression

Conditional logistic regression can also be used for matched data with a binary outcome, however, it is less flexible than mixed effects regression.