TDM 30200: Project 06 - Data Types, Dependency and Mixed Model

What are your first insights from the data so far?

Let’s try to ask some interesting questions, such as "Have players been hitting more Homeruns (HR) over the years? Can you check it over a plot?"

So, start calculating the HR rate (Homeruns per at-bat(AB)) and get a more meaningful measure of how often players hit home runs, independent of season length or other influences.

Note: After you get your plot, you may notice some sharp decreases in HR rates, for example, in 1943. When you do a quick googling, you may see how World War II was effective on it. See this link for more reading.

Add one more question you find interesting from the data and write your own interpretation.

Remember, HR guarantees at least one RBI since the batters score for themselves, and if there are runners on base, the RBI count increases. However, not all RBIs come from HRs; a batter can also earn RBIs through other types of hits, such as singles or doubles, that allow teammates to score. While power hitters who frequently hit HRs tend to have high RBI totals, their RBI count also depends on whether there are runners on base when they come up to bat.

As a result, examining the changes in these two variables individually and together can be important in understanding the hitting power of the players and their contributions to the team. Before looking at these changes, what you will notice in your initial analysis is that since not every player can hit HRs all the time, zero will be dominant in these vectors. Let’s work with values greater than zero for these two variables.

Consider a linear model where you aim to explain changes in the RBI with HR. How effective do you think a simple regression line is in capturing this variability?

Now, let’s take a different approach. This time, we will run a separate linear model for each player. One way to achieve this is by using the lmList function from the lme4 library as follows:

The output of this model gives all parameter estimates for some players, while for others, it gives NaN or NA values. There may be different reasons for this. Before reading the next paragraph, pause for a while and think about what can cause it.

One of the reasons can be that some players might not have enough data points to estimate both the slope (HR coefficient) and intercept. If a player has only one data point, then the regression line is undefined because there is no variation in HR to estimate a slope. If a player has exactly two data points, the regression line is perfectly determined, meaning the standard error cannot be estimated, which can result in NaN.

Additionally, if a player’s HR values do not vary at all, the model cannot compute a meaningful slope, leading to NA values for the coefficient. In cases where RBI and HR have a perfect linear relationship, the regression model may achieve a perfect fit, resulting in undefined or extremely large standard errors. Finally, missing data in either HR or RBI can cause certain players to have too few valid observations, preventing proper coefficient estimation and leading to NA or NaN values in the results.

Here are some more examples:

Some More Visualization Example with Lahman Data:

Now, let’s discuss the previous models a little bit. Explaining this data with a regression model is too optimistic because there is a lot of variability between players. If you examine it in more detail, you will even notice that there is variability between years and leagues. On the other hand, if we model each player, league, or season separately, we introduce a different type of modeling error known as overfitting or loss of generalizability. By fitting a separate model for each subgroup, we risk capturing noise rather than meaningful patterns, making our predictions unreliable for new or unseen data. Additionally, splitting the data this way reduces the number of observations per model, leading to unstable coefficient estimates and high variance in the results. To handle this type of repeated data, different methods will be needed. One approach is to incorporate a random effect, which captures the variability among players within the same model, alongside the regression line.

You can find the following page, built by Michael Freeman, very useful to visually understand the mixed model approach. Please spare five minutes to go over it before you move to the next step: mfviz.com/hierarchical-models/

For the ones who wants to learn more about mixed models, the following books have the downloadable pdf copy:

The mixed model approach will include both your fixed parameter and a subject-specific parameter with random effects. We have the model definition with a random intercept for each player as follows:

$RBI_{ij} = \beta_0 + \beta_1 HR_{ij} + u_{0j} + \epsilon_{ij}$

where:

This formulation allows each player to have their own intercept in a single model while still maintaining an overall trend across all players.

Now, by using the lmer function in the lme4 library, run a mixed model for the previous model. Here is the basic syntax for this:

(1 | playerID): This part adds random effects to your model for each player.

We can see the population (fixef) and player-specific (ranef) effects as follows in R:

Lahman data includes the year variable, which generates the repeats for players, and it also includes a lot of information for modeling. Also, there are a lot of rule changes in baseball. So, it may be worthwhile to add year to account for these changes, too.

Depending on the aim of the research, if you assume that year has a consistent effect across all players, you can add it as a fixed effect. The interpretation for this model can be as follows: the coefficient for the year will show the general trend in RBI over time (e.g., a positive coefficient means RBI totals tend to increase over the years).

Or you may want to add the year as a random effect, which allows each year to have a different baseline effect, capturing year-to-year variability. With this approach, the model accounts for yearly fluctuations in RBI production that aren’t explained by HRs alone.