SMHS LinearModeling LMM

SMHS Linear Modeling - Linear mixed effects analyses

Scientific inference based on fixed and random effect models, assumptions, and mixed effects logistic regression.

Questions:

• What happens if data are not independent and identically distributed (IIDs)?
• How to model multiple observations for the case/subject (across time, conditions, etc.)?

Fixed and random effects Linear models express relationships between data elements (variables) in terms of a (linear) function. For example, we can model weight as a function of height.

Weight ~ Height + ε

Here, “height” a fixed effect, and ε was our “error term” representing the deviations between the model predictions and observations (of weight) due to “random” factors that we cannot control experimentally. This tem (ε) is the “probabilistic” or “stochastic” part of the model. Let’s try to unpack “ε” and add complexity to it. In mixed (fixed and random effect) models, everything in the “systematic” part of your model works just like with linear models. If we change the random aspect of our model this leaves the systematic part (height) unchanged.

Suppose we’re looking at the Baseball data and try to identify a relationship that looks like this:

Weight ~ position + ε

Position (player field position) is treated as a categorical factor with several levels (e.g., Catcher, First-Baseman, etc.) On top of that, we also have an additional fixed effect, Height, and so our bivariate linear model looks more like this:

Weight ~ Height + position + ε

This model expansion is nice, but it complicates a bit the data analytics and scientific inference. If the study design involved taking multiple measures per player, say across time/age, each player would yield multiple position, height and weight responses. According to the assumptions of the linear model, this would violate the independence assumption as multiple responses from the same subject cannot be regarded as independent from one another. Every player has a slightly different weight, and this is going to be an idiosyncratic factor that affects all responses from the same player, thus rendering these different responses inter-dependent (within player) instead of independent, as required by the model assumptions.

A way to resolve this model assumption violation is to add a random effect for players. This allows us to account for inter-independences by assuming a different “baseline” weight value for each player. For example, player 1 may have a mean weight 200 pounds across different times, and player 2 may have a mean weight of 350 pounds. Here’s a visual depiction of how this looks like:

Team A_Player1	Team A_Playe2	Team A_Player3	Team A_Player4	Team A_Player5	Team B_Player1	Team B_Player2	Team B_Player 3
160	340	240	340	180	200	240	180
180	340	240	400	240	200	320	120
220	320	320	400	260	180	340	160
240	300	300	380	320	160	300	160
200	380	300	380	280	260	300	260
220	360	280	360	280	180	300	180
260	360	320	400	260	240	320	180
200	360	280	380	300	220	320	200
220	480	260	380	280	180	300	220

SOCR Charts generated these plots (http://www.socr.umich.edu/html/cha/). The same data may similarly be plotted using R:

data <- read.table('C:\\Users\\Dinov\\Desktop\\01a_data.txt',as.is=T, header=T)
# data <- read.table('https://umich.instructure.com/files/330381/download?download_frd=1&verifier=HpfmjfMFaMsk7rIpfPx0tmz960oTW7JA8ZonGvVC',as.is=T, header=T)	
attach(data)
boxplot(TeamA_Player1, TeamA_Player2, TeamA_Player3, TeamA_Player4, TeamA_Player5, TeamB_Player1, TeamB_Player2, TeamB_Player3, col=c("white","lightgray"))
boxplot(data, las = 2)
boxplot(data, las = 2, par(mar = c(8, 5, 4, 2)+ 0.1))

....

Next See

Machine Learning Algorithms section for data modeling, training , testing, forecasting, prediction, and simulation.

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• SOCR Home page: http://www.socr.umich.edu

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