Teaching materials for Python MixedLM (mixed linear models)
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README.md
nhanes_bp.py

README.md

Mixed Linear Models (MixedLM) in Python Statsmodels

Linear mixed Models

Mixed models are a form of regression model, meaning that the goal is to relate one dependent variable (also known as the outcome or response) to one or more independent variables (known as predictors, covariates, or regressors). Mixed models are typically used when there may be statistical dependencies among the observations. More basic regression procedures like least squares regression and generalized linear models (GLM) take the observations to be independent of each other. Although it is sometimes possible to use OLS or GLM with dependent data, usually an alternative approach that explicitly accounts for any statistical dependencies in the data is a better choice

Terminology: The following terms are mostly equivalent: mixed model, mixed effects model, multilevel model, hierarchical model, random effects model, variance components model.

Alternatives and related approaches: Here we focus on using mixed linear models to capture structural trends and statistical dependencies among data values. Other approaches with related goals include generalized least squares (GLS), generalized estimating equations (GEE), fixed effects regression, and various forms of marginal regression.

Nonlinear mixed models: Here we only consider linear mixed models. Generalized linear mixed models ("GLIMMIX") and non-linear mixed effects models also exist, but are not currently available in Python Statsmodels.

Mean and variance structure

Many regression approaches can be interpreted in terms of the way that they specify the mean structure and the variance structure of the population being modeled. The mean structure can be written as E[Y|X], read as "the mean of Y given X". For example, if your dependent variable is a person's income, and the predictors are their age, number of years of schooling, and gender, you might model the mean structure as

E[income | age, school, female] = b0 + b1⋅age + b2⋅school + b3⋅female.

This is a linear mean structure, which is the mean structure used in linear regression (e.g. OLS), and in linear mixed models. The parameters b0, b1, b2, and b3 are unknown constants to be fit to the data, while income, age, education, and gender are observed data values. The term "linear" here refers to the fact that the mean structure is linear in the parameters (b0, b1, b2, b3). Note that it is not necessary for the mean structure to be linear in the data. For example, we would still have a linear model if we had specified the mean structure as

E[income | age, school, female] = b0 + b1⋅age + b2⋅age^2 + b3⋅school + b4⋅female

The variance structure can be written as Var[Y|X], and is read as "the variance of Y given X". A very basic variance structure is a constant or homoscedastic variance structure. For the income analysis discussed above, this would mean that

Var[income | age, school, female] = v,

where v is an unkown constant to be fit to the data. We will see more complex non-constant variance structures below.

In the context of mixed models, the mean and variance structures are often referred to as the marginal mean structure and marginal variance structure, for reasons that will be explained further below.

Dependent data

A common situation in applied research is that several observations are obtained for each person in a sample. These might be replicates of the same measurement taken at one point in time (e.g. triplicate blood pressure measurements), longitudinal measurements of the same trait taken over time (e.g. annual BMI measurements taken over several years), or related traits measured at the same or different times (e.g. hearing levels in the left and right ear). When data are collected this way, it is likely that the measures within a single person are correlated.

Dependent data often arise when taking repeated measurements on each person, but other sources of dependence are also possible. For example, we may have test scores on students in a classroom, with the classroom nested in a school, which in turn is nested in a school district, etc. In this case, the students in one classroom (or school, etc.) may tend to score higher, or lower than students in other classrooms or schools. This constitutes a form of statistical dependence. The generic terms cluster variable or grouping variable are often used to refer to whatever is the main unit of analysis on which the repeated measures are made (people, classrooms, etc.).

Longitudinal data and random coefficients

There are various ways to accommodate correlations within a regression framework. In mixed modeling, the parameters in the regression model are taken to vary from one cluster to the next. For example, if we have repeated measures of blood pressure and age over time within a person, we can regress blood pressure on age using a conditionally linear mean structure in which the intercept and slope are subject-specific. That is, for subject i we have

E[SBP(age, i) | a(i), b(i)] = a + b⋅age(i) + a(i) + b(i)⋅age(i)

This model states that for each subject, the blood pressure (SBP) trends linearly with age, but each subject has their own intercept a(i) and slope b(i). These subject-specific parameters modify the population parameters a and b, i.e. the overall intercept for subject i is a + a(i), and the overall slope is b + b(i).

The model above is expressed in conditional mean form. Alternatively, it may be expressed as a fully generative model

SBP(age, i) = a + b⋅age + a(i) + b(i)⋅age(i) + e(age, i),

where e(age, i) are independent "errors", which represent unexplained or unstructured variation.

To simplify interpretation, suppose that age is coded as years beyond age 20. Then the varying intercepts a(i) represent variation between people in their blood pressure at age 20, and the varying slopes b(i) represent variation in the rate at which blood pressure changes with respect to age.

In a mixed model, the values of a(i) and b(i) are treated as random values. Each of these terms has a variance, e.g. var[a(i)] = v_a, and var[b(i)] = v_b. Also, there is a covariance c_ab = cov[a(i), b(i)] between the terms.

Here we provide a simple numerical example. Suppose the mean trend is b = 0.6, meaning that an average person's blood pressure increases 6 units (mm Hg) per decade. If the standard deviation sqrt(v_b) of the random slopes is 0.5, this means that everyone has their own slope, with around 16% of the population having a slope greater than 0.6+0.5 = 1.1, and 16% of the population having a slope less than 0.6-0.5 = 0.1 (the random effects are taken to be Gaussian, and we use here that 16% of a Gaussian population falls more than 1 standard deviation from the mean value in one direction).

Marginal covariance structure for longitudinal models

Above we expressed a linear mixed model using conditional equations, relating observed and unobserved values (random effects). To fully specify a model in this way, we need not only these equations, but also expressions defining the variances of the different random effects, and their correlations with each other.

An alternative way to express a linear mixed model is in terms of its marginal mean and variance structure, E[Y|X] and Var[Y|X], as defined above. We can convert the conditional equations to marginal moments with some simple calculations.

Suppose we observe longitdinal data with three time points per person, taken at the same three time points. If we model these points as above, we have

Y(i, t) = a + b*t + a(i) + b(i)*t + e(i, t)

where t=1,2,3. We can directly calculate the mean struture as

Y(i, t) = a + b*t

The variance structure is

Var[Y(i, t)] = v_a + v_bt^2 + 2c_ab*t + s^2,

where s^2 = Var[e(i, t)], and the covariance between two different time points is

Cov[Y(i, s), Y(i, t)] = v_a + c_ab(s + t) + v_bst.

If the three time points are coded t=-1, 0, 1, we can more explicitly write the marginal mean as

( a - b )
( a     )
( a + b )

and the covariance as

( v_a + v_b - 2*c_ab + s^2   v_a - c_ab       v_a - v_b                )
( v_a - c_ab                 v_a + s^2        v_a + c_ab               )
( v_a - v_b                  v_a + c_ab       v_a + v_b + 2*c_ab + s^2 )

Nested variance components

Above we focused on a setting in which the repeated measures are predicted by a quantitative variable (age) and are taken over time within each subject. A different setting is when the repeated measures are taken for various grouping variables that may be nested or crossed. These are often described as "variance components", but can also be called "random intercepts" or simply "random effects".

A simple example would be if we had data on body mass index (BMI) for subjects where we also know their residential location in terms of neighborhood, city, and state. The neighborhoods are nested in the cities, and the cities are nested in the states. It would be possible to approach this analysis using "fixed effects regression", in which we allocate a parameter to each clustering unit (e.g. to each neighborhood).

Rather than estimating a large number of fixed effects parameters, we can focus instead on estimating the variance contributed by each level of the nesting. A model for these data could be

Y(i) = m + N[n(i)] + C[c(i)] + S[s(i)] + e(i)

where Y(i) is the BMI for subject i, m is the population mean, n(i), c(i), and s(i) are, resepectively, the neighborhood, city, and state where subject i lives, and N[⋅], C[⋅], S[⋅] are the random effects for each of these levels. Suppose that subject i lives in the Burns Park neighborhood of Ann Arbor, Michigan. Then n(i) = Burns Park, c(i) = Ann Arbor, and s(i) = Michigan.

There are unkown random terms associated with each of these levels of clustering. For example perhaps N[Burns Park] = 1, C[Ann Arbor] = 2, and S[Michigan] = -1. This means that people in Burns Park tend to have 1 unit higher BMI than other people in Ann Arbor, people in Ann Arbor have 2 units higher BMI than other people in Michigan, and people in Michigan have one unit lower BMI than people in other states. These terms are statistically independent and combine additively, so that subject i has conditional mean value m + 1 + 2 - 1 = m+2.

In variance components modeling, we imagine that all the N[] terms come from a common distribution, say with mean 0 and variance v_N, all the C[] terms come from a distribution with mean 0 and variance v_C, and all the S[] terms come from a distribution with mean 0 and variance v_S. Our only goal here is to estimate v_N, v_C, and v_S, to better understand how the different levels of geography contribute to the observed value of BMI.

Marginal covariance structure for nested model

As above, we can determine the marginal mean and covariance corresponding to the conditionally-specified model above. The mean is imply the unknown constant m. The variances simply add, so the variance of any observation is

Var[Y] = v_N + v_C + v_S + s^2,

where v_N = Var[N], for the neighborhood random effects N, v_C = Var[C], and so on.

The covariance between two observations depends on how many levels of grouping the observations share in common. Since the grouping levels are nested, if two observations are in the same cluster at a given level, they are also in the same level of the higher clusters. For example, two people who live in the same city must also live in the same state.

To put this to use, suppose we have three people, who live in states S1, S1, S2 (i.e. the first two people live in the same state, and the third person lives in a different state), and they live in cities C1, C1, C2 and neighborhoods N1, N2, N3. Then the marginal covariance matrix for these three people is:

( v_N + v_C + v_S + s^2  v_C + v_S              0                     )
( v_C + v_S              v_N + v_C + v_S + s^2  0                     )
( 0                      0                      v_N + v_C + v_S + s^2 )

Crossed variance components

Variance components can also be "crossed", which basically means "not nested". In a crossed model, variance terms for different variables can occur in arbitrary combinations with each other.

Fitting a crossed model puts more stress on the software, but it can be done if there aren't too many levels of crossing. Suppose for example that we have the number of emails sent among people in a sample. For simplicity, we model these counts with a Gaussian distribution so we can use linear mixed models. We can imagine that each person has a propensity A[] to send emails, and also a propensity B[] to receive emails. Higher values of A indicate that a person writes a lot of emails, while higher values of B indicate that a person recieves a lot of emails. In practice, these values may be correalated, but here we model them as if they were independent.

A simple additive variance component would be

Y(i,j) = m + A[i] + B[j] + e(i, j),

where Y(i, j) is the number of emails sent from subject i to subject j. The random effect A[i] reflects person i's propensity to send emails, and B[j] represents person j's propensity to receive emails. These random effects are crossed, meaning that any of the A[] terms can occur in combination with any of the B[] terms. We are mainly interested in the variance parameters v_a and v_b, describing, respectively, the variation in the population of email sending and email receiving propensities.

Marginal covariance structure for crossed models

The crossed model specified above has a very simple mean structure, in which every observation has the same mean m, which is an unknown parameter to be estimated from the data.

The variance is also fairly simple. Every observation has variance

V[Y] = v_a + v_b + s^2.

There are three possible covariances:

The covariance between Y(i, j) and Y(i, k) (i.e. between the counts for the same sender to different receivers) is v_a.

The covariance between Y(i, k) and Y(k, j) (between the counts for the same receiver with two different senders) is v_b.

The covariance between Y(i, j) and Y(k, l) (different receivers and different senders) is 0.

More general model formulations

Above we gave examples of longitudinal, purely nested, and purely crossed mixed models. In general, a mixed model can have any combination of terms of various types. The notion of a mixed model is very broad and there is no formal definition of exactly what scope of models falls into this class. Certain types of time series models, spatial-temporal models, and structural-equation models can be viewed as mixed models, but may not be able to be fit using standard mixed modeling software tools.

Parameters and random effects

Like the widely-used routines in R, Stata, and SAS, Python Statsmodels uses maximum likelihood to fit mixed models to data. (Technically, the default estimator is restricted maximum likelihood, but the difference is not important here). This means that we are optimizing the parameter values in a class of parametric models to best fit the data. The random effects, e.g. random intercepts a(i) or N[] in the examples discussed above, are random variables, not parameters, but unlike the data (which are also treated as random variables), the random effects are not observed. We therefore marginalize the random effects out of the model's likelihood function before fitting to the data. This means that the fitting process does not directly involve these random effects, although it does involve the parameters defining their distribution.

As discussed above, the parameters in a mixed model can broadly be considered as being one of the following types:

  • Mean structure parameters: this includes regression intercepts and slopes. These parameters determine the marginal mean structure (defined above) but are not sufficient to describe the conditional mean structure, which also depends on a subject's random effects. These parameters are sometimes called "fixed effects" because they describe the marginal trends in the population, not the unique trends for individual subjects.

  • Variance structure parameters: this includes variances of random effects, and covariance parameters describing how various random effects are correlated. These are structural parameters describing how the random effects are distributed, not the random effects themselves.

Since the random effects are not parameters, they are not estimated (this is a good thing). However it is possible to predict the value of a random effect after fitting a model. There are various ways to do this, but the most common approach uses a "Best Linear Unbiased Predictor" (BLUP). There is also some controversy over how to interpret these predictions, and how to do statistical inference with them.

In the longitudinal mixed model

E[SBP(age, i) | a(i), b(i)] = a + b⋅age + a(i) + b(i)⋅age(i)

where a and b are mean structure parameters (fixed effects), and v_a, v_b, c_ab, and the "error variance" V[SBP(age, i) | a(i), b(i)] are variance structure parameters. The a(i) and b(i) are the actual random effects.

In the nested variance components model

Y(i) = m + N[n(i)] + C[c(i)] + S[s(i)] + e(i)

the only mean structure parameter is m. The variance structure parameters are v_n, v_c, v_s, and the error variance (in a variance components model the random effects are independent within and between levels, so there are no covariance parameters).

In the crossed variance components model

Y(i,j) = m + A[i] + B[j] + e(i, j),

the mean structure parameter is m, and the variance structure parameters are v_a, v_b, and the error variance (again there are no covariance parameters).

Software and algorithms

Estimation routines for linear mixed models are much more challenging to implement than routines for fitting more basic regression approaches such as OLS and GLM. However a series of developments in the past 20 years has led to algorithms that are reasonably fast and stable. Statsmodels utilizes many of these best practices, such as internally re-parameterizing the covariance parameters through their Cholesky factor, and profiling out certain parameters during the estimation process.

The earlier specifications of linear models (e.g. Laird and Ware 1982) were explicitly group-based. This means that there was a grouping variable such as a person, such that observations made on different groups are taken to be independent. Many applications of mixed modeling are compatible with this group-based approach. However heavily crossed models that are widely used in, for example, experimental psychology and linguistics are not.

Recent versions of R's LMER have taken a somewhat different algorithmic approach, utilizing the sparse Cholesky factorization. Statsmodels does not use this approach, partly because the sparse Cholesky code is not available with a Python-compatible license. The sparse Chleksy approach may be somewhat more efficient for handling large crossed models as noted above. However Python Statsmodels does use sparse matrices and exploits some matrix factorizations to allow crossed models to be fit.

Another important disctinction between Python Statsmodels and LMER in R (which is the most mature open-source implementation of mixed models) is that the Statsmodels code is written in Python, whereas LMER is mostly written in C that is then linked to R. The Python MixedLM code makes use of advanced Numpy and Scipy techniques (which are written in C) and therefore the distinction is not as clear as it may at first seem. There are many trade-offs in this decision, but at present the Python code generally runs somewhat slower than LMER. There are many innovations underway to accelerate numerical Python code so it is likely that the Statsmodels code will become faster over time.

Other practicalities

To fit a mixed model to data using Python Statsmodels (or most other software tools), it should be in "long format". This means that there is one row of data for each observed outcome (not for each group). If the data are originally represented in wide format, like this

Subject   Time1Y  Time2Y  Time1X   Time2X
1         34      39      12        9
2         31      27      19       15
...

then it should be restructured to long form:

Subject   Time    Y     X
1         1       34    12
1         1       39     9
2         2       31    19
2         2       27    15
...

There are various tools for doing this in Python, including many powerful data manipulation routines in the Pandas library.