ALE-based statistics for statistical inference and effect sizes

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Accumulated local effects (ALE) was developed by Daniel Apley and Jingyu Zhu as a global explanation approach for interpretable machine learning (IML). However, the ale package aims to extend it for statistical inference, among other extensions. This vignette presents the initial effort at extending ALE for statistical inference. In particular, we present some effect size measures specific to ALE. We introduce these statistics in detail in a working paper: Okoli, Chitu. 2023. "Statistical Inference Using Machine Learning and Classical Techniques Based on Accumulated Local Effects (ALE)." arXiv. https://doi.org/10.48550/arXiv.2310.09877. Please note that they might be further refined after peer review.

We begin by loading the necessary libraries.

library(mgcv)   # for datasets and the gam function
library(dplyr)  # for data manipulation
library(ale)

Example dataset

We will demonstrate ALE statistics using a dataset composed and transformed from the mgcv package. This package is required to create the generalized additive model (GAM) that we will use for this demonstration. (Strictly speaking, the source datasets are in the nlme package, which is loaded automatically when we load the mgcv package.) Here is the code to generate the data that we will work with:

# Create and prepare the data

# Specific seed chosen to illustrate the spuriousness of the random variable
set.seed(6)  

math <- 
  # Start with math achievement scores per student
  MathAchieve |> 
  as_tibble() |> 
  mutate(
    school = School |> as.character() |>  as.integer(),
    minority = Minority == 'Yes',
    female = Sex == 'Female'
  ) |> 
  # summarize the scores to give per-school values
  summarize(
    .by = school,
    minority_ratio = mean(minority),
    female_ratio = mean(female),
    math_avg = mean(MathAch),
  ) |> 
  # merge the summarized student data with the school data
  inner_join(
    MathAchSchool |> 
      mutate(school = School |> as.character() |>  as.integer()),
    by = c('school' = 'school')
  ) |> 
  mutate(
    public = Sector == 'Public',
    high_minority = HIMINTY == 1,
  ) |> 
  select(-School, -Sector, -HIMINTY) |> 
  rename(
    size = Size,
    academic_ratio = PRACAD,
    discrim = DISCLIM,
    mean_ses = MEANSES,
  ) |> 
  # Remove ID column for analysis
  select(-school) |> 
  select(
    math_avg, size, public, academic_ratio,
    female_ratio, mean_ses, minority_ratio, high_minority, discrim,
    everything()
  ) |> 
  mutate(
    rand_norm = rnorm(nrow(MathAchSchool)) 
  )

glimpse(math)

The structure has 160 rows, each of which refers to a school whose students have taken a mathematics achievement test. We describe the data here based on documentation from the nlme package but many details are not quite clear:

| variable | format | description | |------------------------------|------------------|------------------------| | math_avg | double | average mathematics achievement scores of all students in the school | | size | double | the number of students in the school | | public | logical | TRUE if the school is in the public sector; FALSE if in the Catholic sector | | academic_ratio | double | the percentage of students on the academic track | | female_ratio | double | percentage of students in the school that are female | | mean_ses | double | mean socioeconomic status for the students in the school (measurement is not quite clear) | | minority_ratio | double | percentage of students that are members of a minority racial group | | high_minority | logical | TRUE if the school has a high ratio of students of minority racial groups (unclear, but perhaps relative to the location of the school) | | discrim | double | the "discrimination climate" (perhaps an indication of extent of racial discrimination in the school?) | | rand_norm | double | a completely random variable |

Of particular note is the variable rand_norm. We have added this completely random variable (with a normal distribution) to demonstrate what randomness looks like in our analysis. (However, we selected the specific random seed of 6 because it highlights some particularly interesting points.)

The outcome variable that is the focus of our analysis is math_avg, the average mathematics achievement scores of all students in each school. Here are its descriptive statistics:

summary(math$math_avg)

Enable progress bars

Before starting, we recommend that you enable progress bars to see how long procedures will take. Simply run the following code at the beginning of your R session:

# Run this in an R console; it will not work directly within an R Markdown or Quarto block
progressr::handlers(global = TRUE)
progressr::handlers('cli')

If you forget to do that, the {ale} package will do it automatically for you with a notification message.

Full model bootstrap

Now we create a model and compute statistics on it. Because this is a relatively small dataset, we will carry out full model bootstrapping using the model_bootstrap() function.

First, we create a generalized additive model (GAM) so that we can capture non-linear relationships in the data.

gam_math <- gam(
     math_avg ~ public + high_minority +
     s(size) + s(academic_ratio) + s(female_ratio) + s(mean_ses) + 
     s(minority_ratio) + s(discrim) + s(rand_norm),
     data = math
   )

gam_math

Create p-value function objects

Before we bootstrap the model to create ALE and other data, there is an important preliminary step when our goal is to analyze ALE statistics. As with any statistics calculated on a dataset, there is some randomness to the statistic values that the procedure will give us. To quantify this randomness, we want to obtain p-values for these statistics. A p-value is a number from 0 to 1 that indicates the probability that a given statistic value would occur by random chance. So, high p-values mean that the statistic value is likely to be random whereas low p-values (typically lower than 0.05 by convention) mean that the statistic probably represents a reliable value, not obtained merely by chance. (This is not to be confused with effect sizes, which we come to later in this article.)

P-values for most standard statistics are based on the assumption that these statistics fit some distribution or another (e.g., Student's t, $\chi^2$, etc.). With these distributional assumptions, p-values can be calculated very quickly. However, a key characteristic of ALE is that there are no distributional assumptions: ALE data is a description of a model's characterization of the data given to it. Accordingly, ALE statistics do not assume any distribution, either.

The implication of this for p-values is that the distribution of the data must be discovered by simulation rather than calculated based on distributional assumptions. The procedure for calculating p-values is the following:

As you can imagine, this procedure is very slow: it involves retraining the entire model on the full dataset 1,000 times. The {ale} package speeds of the process significantly through parallel processing (implemented by default), but it still involves the speed of retraining the model hundreds of times.

To avoid having to repeat this procedure several times (as would be the case when you are doing exploratory analyses), the create_p_funs() function generates a p_funs object that can be run once for a given model-dataset pair. The p_funs object contains functions that can generate p-values based on statistics for any variable from the same model-dataset pair. It generates these p-values when passed to the ale() or model_bootstrap() functions. For very large datasets, the process of generating the p_funs object could be sped up by only using a subset of the data and by running fewer than 1,000 random iterations by setting the rand_it argument. However, the create_p_funs() function will not allow fewer than 100 iterations, otherwise the p-values thus generated would be meaningless.)

We now demonstrate how to create the p_funs object for our case.

# # To generate the code, uncomment the following lines.
# # But it is slow because it retrains the model 1000 times,
# # so this vignette loads a pre-created p-values object.
# gam_math_p_funs <- create_p_funs(
#   math,
#   gam_math
# )
# saveRDS(gam_math_p_funs, file.choose())
gam_math_p_funs <- url('https://github.com/tripartio/ale/raw/main/download/gam_math_p_funs.rds') |> 
  readRDS()

We can now proceed to bootstrap the model for ALE analysis.

Bootstrap the model with p-values

By default, model_bootstrap() runs 100 bootstrap iterations; this can be controlled with the boot_it argument. Bootstrapping is usually rather slow, even on small datasets, since the entire process is repeated that many times. The model_bootstrap() function speeds of the process significantly through parallel processing (implemented by default), but it still involves retraining the entire model dozens of times. The default of 100 should be sufficiently stable for model building, when you would want to run the bootstrapped algorithm several times and you do not want it to be too slow each time. For definitive conclusions, you could run 1,000 bootstraps or more to confirm the results of 100 bootstraps.

mb_gam_math <- model_bootstrap(
  math, 
  gam_math,
  # Pass the p_funs object so that p-values will be generated
  ale_options = list(p_values = gam_math_p_funs),
  # For the GAM model coefficients, show details of all variables, parametric or not
  tidy_options = list(parametric = TRUE),
  # tidy_options = list(parametric = NULL),
  boot_it = 40,  # 100 by default but reduced here for a faster demonstration
  parallel = 2  # CRAN limit (delete this line on your own computer for faster speed)
)

We can see the bootstrapped values of various overall model statistics by printing the model_stats element of the model bootstrap object:

mb_gam_math$model_stats

The names of the columns follow the broom package conventions:

Our focus, however, in this vignette is on the effects of individual variables. These are available in the model_coefs element of the model bootstrap object:

mb_gam_math$model_coefs

In this vignette, we cannot go into the details of how GAM models work (you can learn more with Noam Ross's excellent tutorial). However, for our model illustration here, the estimates for the parametric variables (the non-numeric ones in our model) are interpreted as regular statistical regression coefficients whereas the estimates for the non-parametric smoothed variables (those whose variable names are encapsulated by the smooth s() function) are actually estimates for expected degrees of freedom (EDF in GAM). The smooth function s() lets GAM model these numeric variables as flexible curves that fit the data better than a straight line. The estimate values for the smooth variables above are not so straightforward to interpret, but suffice it to say that they are completely different from regular regression coefficients.

The ale package uses bootstrap-based confidence intervals, not p-values that assume predetermined distributions, to determine statistical significance. Although they are not quite as simple to interpret as counting the number of stars next to a p-value, they are not that complicated, either. Based on the default 95% confidence intervals, a coefficient is statistically significant if conf.low and conf.high are both positive or both negative. We can filter the results on this criterion:

mb_gam_math$model_coefs |> 
  # filter is TRUE if conf.low and conf.high are both positive or both negative because
  # multiplying two numbers of the same sign results in a positive number.
  filter((conf.low * conf.high) > 0)

The statistical significance of the estimate (EDF) of the smooth terms is meaningless here because EDF cannot go below 1.0. Thus, even the random term s(rand_norm) appears to be "statistically significant". Only the values for the non-smooth (parametric terms) public and high_minority should be considered here. So, we find that neither of the coefficient estimates of public nor of high_minority has an effect that is statistically significantly different from zero. (The intercept is not conceptually meaningful here; it is a statistical artifact.)

This initial analysis highlights two limitations of classical hypothesis-testing analysis. First, it might work suitably well when we use models that have traditional linear regression coefficients. But once we use more advanced models like GAM that flexibly fit the data, we cannot interpret coefficients meaningfully and so it is not so clear how to reach inferential conclusions. Second, a basic challenge with models that are based on the general linear model (including GAM and almost all other statistical analyses) is that their coefficient significance compares the estimates with the null hypothesis that there is no effect. However, even if there is an effect, it might not be practically meaningful. As we will see, ALE-based statistics are explicitly tailored to emphasize practical implications beyond the notion of "statistical significance".

ALE effect size measures

ALE was developed to graphically display the relationship between predictor variables in a model and the outcome regardless of the nature of the model. Thus, before we proceed to describe our extension of effect size measures based on ALE, let us first briefly examine the ALE plots for each variable.

ALE plots with p-values

mb_gam_math$ale$plots |> 
  patchwork::wrap_plots(ncol = 2)

We can see that most variables seem to have some sort of mean effect across various values. However, for statistical inference, our focus must be on the bootstrap intervals. Crucial to our interpretation is the middle grey band that indicates the median ± 5% of random values. Below, we will explain what exactly the ALE range (ALER) means, but for now, we can say this:

The idea is that if the ALE values of any predictor variable falls fully within the ALER band, then it has no greater effect than 95% of purely random variables. Moreover, to consider any effect in the ALE plot to be statistically significant (that is, non-random), there should be no overlap between the bootstrapped confidence regions of a predictor variable and the ALER band. (For the threshold p-values, We use the conventional defaults of 0.05 for 95% confidence and 0.01 for 99% confidence, but the value can be changed with the p_alpha argument.)

For categorical variables (public and high_minority above), the confidence interval bars for all categories overlap the ALER band. The confidence interval bars indicate two useful pieces of information to us. When we compare them to the ALER band, their overlap or lack thereof tells us about the practical significance of the category. When we compare the confidence bars of one category with those of others, it allows us to assess if the category has a statistically significant effect that is different from that of the other categories; this is equivalent to the regular interpretation of coefficients for GAM and other GLM models. In both cases, the confidence interval bars of the TRUE and FALSE categories overlap each other, indicating that there is no statistically significant difference between categories. Whereas the coefficient table above based on classic statistics indicated this conclusion for public, it indicated that high_minority had a statistically significant effect; our ALE analysis indicates that high_minority does not. In addition, each confidence interval band overlaps the ALER band, indicating that none of the effects is meaningfully different from random results, either.

For numeric variables, the confidence regions overlap the ALER band for most of the domains of the predictor variables except for some regions that we will examine. The extreme points of each variable (except for discrim and female_ratio) are usually either slightly below or slightly above the ALER band, indicating that extreme values have the most extreme effects: math achievement increases with increasing school size, academic track ratio, and mean socioeconomic status, whereas it decreases with increasing minority ratio. The ratio of females and the discrimination climate both overlap the ALER band for the entirety of their domains, so any apparent trends are not supported by the data.

Of particular interest is the random variable rand_norm, whose average ALE appears to show some sort of pattern. However, we note that the 95% confidence intervals we use mean that if we were to retry the analysis for twenty different random seeds, we would expect at least one of the random variables to partially escape the bounds of the ALER band. We will return below to the implications of random variables in ALE analysis.

ALE plots without p-values

Before we continue, let us take a brief detour to see what we get if we run model_bootstrap() without passing it a p_funs object. This might be because we forget to do so, or because we want to see quick results without the slow process of first generating a p_funs object. Let us run model_bootstrap() again, but this time, without p-values.

mb_gam_no_p <- model_bootstrap(
  math, 
  gam_math,
  # For the GAM model coefficients, show details of all variables, parametric or not
  tidy_options = list(parametric = TRUE),
  # tidy_options = list(parametric = NULL),
  boot_it = 40,  # 100 by default but reduced here for a faster demonstration
  parallel = 2  # CRAN limit (delete this line on your own computer)
)

mb_gam_no_p$ale$plots |> 
  patchwork::wrap_plots(ncol = 2)

In the absence of p-values, the {ale} packages uses alternate visualizations to offer meaningful results, but with somewhat different interpretations of the middle grey band. Without p-values, we do not have any point of reference for the ALER statistics, so we use percentiles around the median as the reference. The middle grey band here indicates the median ± 2.5%, that is, the middle 5% of all average mathematics achievement scores (math_avg) values in the dataset. We call this the "median band". The idea is that if any predictor can do no better than influencing math_avg to fall within this middle median band, then it only has a minimal effect. For an effect to be considered statistically significant, there should be no overlap between the confidence regions of a predictor variable and the median band. (We use 5% around the median by default, but the value can be changed with the median_band_pct argument.) For further reference, the outer dashed lines indicate the interquartile range of the outcome values, that is, the 25th and 75th percentiles.

We can see that in this case, the 5% median band is much narrower than the 5% ALER band when p-values are calculated, though they might be more similar for a different dataset. This should give us pause to skipping the calculation of p-values, since we might be overly lax in interpreting apparent relationships as meaningful whereas the ALER band indicates that they might not be that different from what random variables might produce.

For most of the rest of this article, we will only analyze the results with ALER bands generated from p-values, though we will briefly revisit median bands without p-values.

ALE effect size measures on the scale of the y outcome variable

Although ALE plots allow rapid and intuitive conclusions for statistical inference, it is often helpful to have summary numbers that quantify the average strengths of the effects of a variable. Thus, we have developed a collection of effect size measures based on ALE tailored for intuitive interpretation. To understand the intuition underlying the various ALE effect size measures, it is useful to first examine the ALE effects plot, which graphically summarizes the effect sizes of all the variables in the ALE analysis. This is generated when ale is executed and both statistics and plots are requested (which is the case by default) and is accessible with the To focus on all the measures for a specific variable, we can access the ale$stats$effects_plot element:

mb_gam_math$ale$stats$effects_plot

This plot is unusual, so it requires some explanation:

Although it is somewhat confusing to have two axes, the percentiles are a direct transformation of the raw outcome values. The first two base ALE effect size measures below are in units of the outcome variable while their normalized versions are in percentiles of the outcome. Thus, the same plot can display the two kinds of measures simultaneously. Referring to this plot can help understand each of the measures, which we proceed to explain in detail.

Before we explain these measures in detail, we must reiterate the timeless reminder that correlation is not causation. So, none of the scores necessarily means that an x variable causes a certain effect on the y outcome; we can only say that the ALE effect size measures indicate associated or related variations between the two variables.

ALE range (ALER)

The easiest ALE statistic to understand is the ALE range (ALER), so we begin there. It is simply the range from the minimum to the maximum of any ale_y value for that variable. Mathematically, that is

$$\mathrm{ALER}(\mathrm{ale_y}) = { \min(\mathrm{ale_y}), \max(\mathrm{ale_y}) }$$\ where $\mathrm{ale_y}$ is the vector of ALE y values for a variable.

All the ALE effect size measures are centred on zero so that they are consistent regardless of if the user chooses to centre their plots on zero, the median, or the mean. Specifically,

ALER shows the extreme values of a variable's effect on the outcome. In the effects plot above, it is indicated by the extreme ends of the horizontal bars for each variable. We can access ALE effect size measures through the ale$stats element of the bootstrap result object, with multiple views. To focus on all the measures for a specific variable, we can access the ale$stats$by_term element.

Let's focus on public. Here is its ALE plot:

mb_gam_math$ale$plots$public

Here are the effect size measures for the categorical public:

mb_gam_math$ale$stats$by_term$public

We see there that public has an ALER of [r c(mb_gam_math$ale$stats$by_term$public$estimate['aler_min'], mb_gam_math$ale$stats$by_term$public$estimate['aler_max']) |> round(2)]. When we consider that the median math score in the dataset is r math$math_avg |> median() |> round(1), this ALER indicates that the minimum of any ALE y value for public (when public == TRUE) is r mb_gam_math$ale$stats$by_term$public$estimate['aler_min'] |> round(2) below the median. This is shown at the r (math$math_avg |> median() + mb_gam_math$ale$stats$by_term$public$estimate['aler_min']) |> round(1) mark in the plot above. The maximum (public == FALSE) is r mb_gam_math$ale$stats$by_term$public$estimate['aler_max'] |> round(2) above the median, shown at the r (math$math_avg |> median() + mb_gam_math$ale$stats$by_term$public$estimate['aler_max']) |> round(1) point above.

The unit for ALER is the same unit as the outcome variable; in our case, that is math_avg ranging from r round(min(math$math_avg, 2)) to r round(max(math$math_avg, 2)). No matter what the average ALE values might be, the ALER quickly shows the minimum and maximum effects of any value of the x variable on the y variable.

For contrast, let us look at a numeric variable, academic_ratio:

mb_gam_math$ale$plots$academic_ratio

Here are its ALE effect size measures:

mb_gam_math$ale$stats$by_term$academic_ratio

The ALER for academic_ratio is considerably broader with r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['aler_min'] |> round(2) below and r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['aler_max'] |> round(2) above the median.

ALE deviation (ALED)

While the ALE range shows the most extreme effects a variable might have on the outcome, the ALE deviation indicates its average effect over its full domain of values. With the zero-centred ALE values, it is conceptually similar to the weighted mean absolute error (MAE) of the ALE y values. Mathematically, it is

$$ \mathrm{ALED}(\mathrm{ale_y}, \mathrm{ale_n}) = \frac{\sum_{i=1}^{k} \left| \mathrm{ale_y}i \times \mathrm{ale_n}_i \right|}{\sum{i=1}^{k} \mathrm{ale_n}_i} $$ where $i$ is the index of $k$ ALE x intervals for the variable (for a categorical variable, this is the number of distinct categories), $\mathrm{ale_y}_i$ is the ALE y value for the $i$th ALE x interval, and $\mathrm{ale_n}_i$ is the number of rows of data in the $i$th ALE x interval.

Based on its ALED, we can say that the average effect on math scores of whether a school is in the public or Catholic sector is r mb_gam_math$ale$stats$by_term$public$estimate['aled'] |> round(2) (again, out of a range from r round(min(math$math_avg, 2)) to r round(max(math$math_avg, 2))). In the effects plot above, the ALED is indicated by a white box bounded by parentheses ( and ). As it is centred on the median, we can readily see that the average effect of school sector barely exceeds the limits of the ALER band, indicating that it barely exceeds our threshold of practical relevance. The average effect for ratio of academic track students is slightly higher at r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['aled'] |> round(2). We can see on the plot that it slightly exceeds the ALER band on both sides, indicating its slightly stronger effect. We will comment on the values of other variables when we discuss the normalized versions of these scores, to which we proceed next.

Normalized ALE effect size measures

Since ALER and ALED scores are scaled on the range of y for a given dataset, these scores cannot be compared across datasets. Thus, we present normalized versions of each with intuitive, comparable values. For intuitive interpretation, we normalize the scores on the minimum, median, and maximum of any dataset. In principle, we divide the zero-centred y values in a dataset into two halves: the lower half from the 0th to the 50th percentile (the median) and the upper half from the 50th to the 100th percentile. (Note that the median is included in both halves). With zero-centred ALE y values, all negative and zero values are converted to their percentile score relative to the lower half of the original y values while all positive ALE y values are converted to their percentile score relative to the upper half. (Technically, this percentile assignment is called the empirical cumulative distribution function (ECDF) of each half.) Each half is then divided by two to scale them from 0 to 50 so that together they can represent 100 percentiles. (Note: when a centred ALE y value of exactly 0 occurs, we choose to include the score of zero ALE y in the lower half because it is analogous to the 50th percentile of all values, which more intuitively belongs in the lower half of 100 percentiles.) The transformed maximum ALE y is then scaled as a percentile from 0 to 100%.

There is a notable complication. This normalization smoothly distributes ALE y values when there are many distinct values, but when there are only a few distinct ALE y values, then even a minimal ALE y deviation can have a relatively large percentile difference. If any ALE y value is less than the difference between the median in the data and the value either immediately below or above the median, we consider that it has virtually no effect. Thus, the normalization sets such minimal ALE y values as zero.

Its formula is:

$$ norm_ale_y = 100 \times \begin{cases} 0 & \text{if } \max(centred_y < 0) \leq ale_y \leq \min(centred_y > 0), \ \frac{-ECDF_{y_{\leq 0}}(ale_y)}{2} & \text{if }ale_y < 0 \ \frac{ECDF_{y_{\geq 0}}(ale_y)}{2} & \text{if }ale_y > 0 \ \end{cases} $$ where - $centred_y$ is the vector of y values centred on the median (that is, the median is subtracted from all values). - $ECDF_{y_{\geq 0}}$ is the ECDF of the non-negative values in y. - $-ECDF_{y_{\leq 0}}$ is the ECDF of the negative values in y after they have been inverted (multiplied by -1).

Of course, the formula could be simplified by multiplying by 50 instead of by 100 and not dividing the ECDFs by two each. But we prefer the form we have given because it is explicit that each ECDF represents only half the percentile range and that the result is scored to 100 percentiles.

Normalized ALER (NALER)

Based on this normalization, we first have the normalized ALER (NALER), which scales the minimum and maximum ALE y values from -50% to +50%, centred on 0%, which represents the median:

$$ \mathrm{NALER}(\mathrm{y, ale_y}) = {\min(\mathrm{norm_ale_y}) + 50, \max(\mathrm{norm_ale_y}) + 50 } $$

where $y$ is the full vector of y values in the original dataset, required to calculate $\mathrm{norm_ale_y}$.

ALER shows the extreme values of a variable's effect on the outcome. In the effects plot above, it is indicated by the extreme ends of the horizontal bars for each variable. We see there that public has an ALER of r c(mb_gam_math$ale$stats$by_term$public$estimate['aler_min'], mb_gam_math$ale$stats$by_term$public$estimate['aler_max']) |> round(2). When we consider that the median math score in the dataset is r math$math_avg |> median() |> round(1), this ALER indicates that the minimum of any ALE y value for public (when public == TRUE) is r mb_gam_math$ale$stats$by_term$public$estimate['aler_min'] |> round(2) below the median. This is shown at the r (math$math_avg |> median() + mb_gam_math$ale$stats$by_term$public$estimate['aler_min']) |> round(1) mark in the plot above. The maximum (public == FALSE) is r mb_gam_math$ale$stats$by_term$public$estimate['aler_max'] |> round(2) above the median, shown at the r (math$math_avg |> median() + mb_gam_math$ale$stats$by_term$public$estimate['aler_max']) |> round(1) point above. The ALER for academic_ratio is considerably broader with r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['aler_min'] |> round(2) below and r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['aler_max'] |> round(2) above the median.

The result of this transformation is that NALER values can be interpreted as percentile effects of y below or above the median, which is centred at 0%. Their numbers represent the limits of the effect of the x variable with units in percentile scores of y. In the effects plot above, because the percentile scale on the top corresponds exactly to the raw scale below, the NALER limits are represented by exactly the same points as the ALER limits; only the scale changes. The scale for ALER and ALED is the lower scale of the raw outcomes; the scale for NALER and NALED is the upper scale of percentiles.

So, with a NALER of r c(mb_gam_math$ale$stats$by_term$public$estimate['naler_min'], mb_gam_math$ale$stats$by_term$public$estimate['naler_max']) |> round(2), the minimum of any ALE value for public (public == TRUE) shifts math scores by r mb_gam_math$ale$stats$by_term$public$estimate['naler_min'] |> round() percentile y points whereas the maximum (public == FALSE) shifts math scores by r mb_gam_math$ale$stats$by_term$public$estimate['naler_max'] |> round() percentile points. Academic track ratio has a NALER of r c(mb_gam_math$ale$stats$by_term$academic_ratio$estimate['naler_min'], mb_gam_math$ale$stats$by_term$academic_ratio$estimate['naler_max']) |> round(2), ranging from r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['naler_min'] |> round() to r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['naler_max'] |> round() percentile points of math scores.

Normalized ALED (NALED)

The normalization of ALED scores applies the same ALED formula as before but on the normalized ALE values instead of on the original ALE y values:

$$ \mathrm{NALED}(y, \mathrm{ale_y}, \mathrm{ale_n}) = \mathrm{ALED}(\mathrm{norm_ale_y}, \mathrm{ale_n}) $$

NALED produces a score that ranges from 0 to 100%. It is essentially the ALED expressed in percentiles, that is, the average effect of a variable over its full domain of values. So, the NALED of public school status of r mb_gam_math$ale$stats$by_term$public$estimate['naled'] |> round(1) indicates that its average effect on math scores spans the middle r mb_gam_math$ale$stats$by_term$public$estimate['naled'] |> round(1) percent of scores. Academic ratio has an average effect expressed in NALED of r mb_gam_math$ale$stats$by_term$academic_ratio$estimate['naled'] |> round(1)% of scores.

The median band and random variables

When we do not have p-values, the NALED is particularly helpful in comparing the practical relevance of variables against our threshold for the median band by which we consider that a variable needs to shift the outcome on average by more than 5% of the median values. This threshold is the same scale as the NALED. So, we can tell that public school status with its NALED of r mb_gam_math$ale$stats$by_term$public$estimate['naled'] |> round(1) just barely crosses our threshold.

It is particularly striking to note the ALE effect size measures for the random rand_norm:

mb_gam_math$ale$plots$rand_norm
mb_gam_math$ale$stats$by_term$rand_norm

rand_norm has a NALED of r mb_gam_math$ale$stats$by_term$rand_norm$estimate['naled'] |> round(1). It might be surprising that a purely random value has any "effect size" to speak of, but statistically, it must have some numeric value or the other. However, by setting our default value for the median band at 5%, we effectively exclude rand_norm from serious consideration. In informal tests with several different random seeds, the random variables never exceeded this 5% threshold. Setting the median band too low at a value like 1% would not have excluded the random variable, but 5% seems like a nice balance. Thus, the effect of a variable like the discrimination climate score (discrim, r mb_gam_math$ale$stats$by_term$discrim$estimate['naled'] |> round(1)) should probably not be considered practically meaningful.

We realize that 5% as a threshold for the median band is rather arbitrary, inspired by traditional $\alpha$ = 0.05 for statistical significance and confidence intervals. A proper analysis should use p-values, as most of this article does. However, our initial analyses here show that 5% seems to be an effective choice for excluding a purely random variable from consideration, even for quick initial analyses.

We return to using p-values for the rest of this article.

Interpretation of normalized ALE effect sizes

Here we summarize some general principles for interpreting normalized ALE effect sizes.

In general, regardless of the values of ALE statistics, we should always visually inspect the ALE plots to identify and interpret patterns of relationships between inputs and the outcome.

A common question for interpreting effect sizes is, "How strong does an effect need to be to be considered 'strong' or 'weak'?" On one hand, we refuse to offer general guidelines for how "strong" is "strong". The simple answer is that it depends entirely on the applied context. It is not meaningful to try to propose numerical values for statistics that are supposed to be useful for all applied contexts.

On the other hand, we do consider it very important to delineate the threshold between random effects and non-random effects. It is always important to distinguish between a weak but real effect from one that is just a statistical artifact due to random chance. For that, we can offer some general guidelines based on whether or not we have p-values.

When we have p-values for ALE statistics, then the boundaries of ALER should generally be used to determine the acceptable risk of considering a statistic to be meaningful. Statistically significant ALE effects are those that are less than the 0.05 p-value ALER minimum of a random variable and greater than the 0.05 p-value maximum of a random variable. As we explained above when introducing the ALER band, this is precisely what the {ale} package does, especially in the plots that highlight the ALER band and the confidence region tables that use the specified ALER p-value threshold.

In the absence of p-values, we suggest that NALED can be a general guide for non-random values. In our informal tests, we find that NALED values below 5% have the same average effect as a random variable. That is, the average effect is not reliable; it might be random. However, regardless of the average effect indicated by NALED, large NALER effects indicate that the ALE plot should be inspected to interpret the exceptional cases. This caveat is very important; unlike GLM coefficients, ALE analysis is sensitive to exceptions to the overall trend. This is precisely what makes it valuable for detecting non-linear effects.

In general, if NALED \< 5%, NALER minimum > –5%, and NALER maximum \< +5%, the input variable has no meaningful effect. All other cases are worth inspecting the ALE plots for careful interpretation: - NALED > 5% means a meaningful average effect. - NALER minimum \< –5% means that there might be at least one input value that significantly lowers the outcome values. - NALER maximum > +5% means that there might be at least one input value that significantly increases the outcome values.

Statistical inference with ALE

Although effect sizes are valuable in summarizing the global effects of each variable, they mask much nuance since each variable varies in its effect along its domain of values. Thus, ALE is particularly powerful in its ability to make fine-grained inferences of a variable's effect depending on its specific value.

ALE data structures for categorical and numeric variables

To understand how bootstrapped ALE can be used for statistical inference, we must understand the structure of ALE data. Let's begin simple with a binary variable with just two categories, public:

mb_gam_math$ale$data$public

Here is the meaning of each column of ale$data for a categorical variable:

By default, the ale package centres ALE values on the median of the outcome variable; in our dataset, the median of all the schools' average mathematics achievement scores is r math$math_avg |> median() |> round(1). With ALE centred on the median, the weighted sum of ALE y values (weighted on ale_n) above the median is approximately equal to the weighted sum of those below the median. So, in the ALE plots above, when you consider the number of instances indicated by the rug plots and category percentages, the average weighted ALE y approximately equals the median.

Here is the ALE data structure for a numeric variable, academic_ratio:

mb_gam_math$ale$data$academic_ratio

The columns are the same as with a categorical variable, but the meaning of ale_x is different since there are no categories. To calculate ALE for numeric variables, the range of x values is divided into fixed intervals (by default 100, customizable with the x_intervals argument). If the x values have fewer than 100 distinct values in the data, then each distinct value becomes an ale_x interval. (This is often the case with smaller datasets like ours; here academic_ratio has only 65 distinct values.) If there are more than 100 distinct values, then the range is divided into 100 percentile groups. So, ale_x represents each of these x-variable intervals. The other columns mean the same thing as with categorical variables: ale_n is the number of rows of data in each ale_x interval and ale_y is the calculated ALE for that ale_x value.

Bootstrap-based inference with ALE

In a bootstrapped ALE plot, values within the confidence intervals are statistically significant; values outside of the ALER band can be considered at least somewhat meaningful. Thus, the essence of ALE-based statistical inference is that only effects that are simultaneously within the confidence intervals AND outside of the ALER band should be considered conceptually meaningful.

We can see this, for example, with the plot of mean_ses:

mb_gam_math$ale$plots$mean_ses

It might not always be easy to tell from a plot which regions are relevant, so the results of statistical significance are summarized with the ale$conf_regions$by_term element, which can be accessed for each variable from its by_term element:

mb_gam_math$ale$conf_regions$by_term$mean_ses

For numeric variables, the confidence regions summary has one row for each consecutive sequence of x values that have the same status: all values in the region are below the middle irrelevance band, they overlap the band, or they are all above the band. Here are the summary components:

These results tell us that, for mean_ses, from -1.19 to -1.04, ALE is below the median band from 6.1 to 7.6. From -0.792 to -0.792, ALE overlaps the median band from 10.2 to 10.2. From -0.756 to -0.674, ALE is below the median band from 10.2 to 10.8. From -0.663 to -0.663, ALE overlaps the median band from 10.9 to 10.9. From -0.643 to -0.484, ALE is below the median band from 10.8 to 11.2. From -0.467 to -0.467, ALE overlaps the median band from 11.5 to 11.5. From -0.46 to -0.46, ALE is below the median band from 11.4 to 11.4. A few other regions briefly exceeded the ALER band.-

Interestingly, most of the text of the previous paragraph was generated automatically by an internal (unexported function) ale:::summarize_conf_regions_in_words. (Since the function is not exported, you must use ale::: with three colons, not just two, if you want to access it.)

ale:::summarize_conf_regions_in_words(mb_gam_math$ale$conf_regions$by_term$mean_ses)

While the wording is rather mechanical, it nonetheless illustrates the potential value of being able to summarize the inferentially relevant conclusions in tabular form.

Confidence region summary tables are available not only for numeric but also for categorical variables, as we see with public. Here is its ALE plot again:

mb_gam_math$ale$plots$public

And here is its confidence regions summary table:

mb_gam_math$ale$conf_regions$by_term$public

Since we have categories here, there is no start or end positions and there is no trend. We instead have each x category and its single ALE y value, with the n and n_pct of the respective category and relative_to_mid as before to indicate whether the indicated category is below, overlaps with, or is above the ALER band.

Again with the help of ale:::summarize_conf_regions_in_words, these results tell us that, for public, for FALSE, the ALE of 13.3 overlaps the ALER band. For TRUE, the ALE of 12.6 overlaps the ALER band.

Again, our random variable rand_norm is particularly interesting. Here is its ALE plot:

mb_gam_math$ale$plots$rand_norm

And here is its confidence regions summary table:

mb_gam_math$ale$conf_regions$by_term$rand_norm

Despite any apparent pattern, we see that from -2.4 to 2.61, ALE overlaps the median band from 12 to 12.8. So, despite the random highs and lows in the bootstrap confidence interval, there is no reason to suppose that the random variable has any effect anywhere in its domain.

We can conveniently summarize all the confidence regions from all variables that are statistically significant or meaningful by accessing the conf_regions$significant element:

mb_gam_math$ale$conf_regions$significant

This summary focuses only on the x variables that have meaningful ALE regions anywhere in their domain. We can also conveniently isolate which variables have any such meaningful region by extracting the unique values in the term column:

mb_gam_math$ale$conf_regions$significant$term |> 
  unique()

This is especially useful for analyses with dozens of variables; we can thus quickly isolate and focus on the most meaningful ones.



Try the ale package in your browser

Any scripts or data that you put into this service are public.

ale documentation built on May 29, 2024, 10:33 a.m.