One of the many subtasks when performing a prediction scheme is adjusting for mediating and confounding variables. But what is a mediator, what is a confounder, and why include thhem? Counfounder and Mediators can be illustrated using a causal diagram (From this Wikipedia page):
As you can see, a confounding variable is something that confounds or ‘confuses’ the relationship between an exposure and an outcome. The relationship between an exposure X and an outcome Y is influenced by a confounding variable Z. A mediating variable connects an exposure to an outcome where otherwise would not be. The relationship between an exposure X and an outcome Y is influenced by a mediating variable Z.
Here, I will show how MCCV can be used to identify confounders and mediators.
Generate dataset with an effect X, outcome Y, and a confounder Z variables
The random variable Z generates X and Y. Any correlation between X and Y is therefore spurious
import numpy as np
N=100
Z1 = np.random.normal(loc=0,scale=1,size=N)
Z2 = np.random.normal(loc=1,scale=1,size=N)
Z = np.concatenate([Z1,Z2])
import scipy as sc
Y = np.concatenate([sc.stats.bernoulli.rvs(z,size=1) for z in (Z - min(Z)) / (max(Z) - min(Z))])
X = Z + np.random.normal(loc=0,scale=1,size=len(Z))df <- tibble::tibble(
X = reticulate::py$X,
Y = reticulate::py$Y,
Z = reticulate::py$Z
)
df[['Y']] <- factor(df$Y,levels=c(0,1))
df# A tibble: 200 × 3
X Y Z
<dbl[1d]> <fct> <dbl[1d]>
1 -0.861 0 -0.0394
2 -0.0157 0 -0.649
3 0.857 0 0.626
4 1.78 0 1.84
5 -1.33 0 -0.539
6 0.688 0 -1.40
7 -0.132 0 0.949
8 0.636 0 -0.702
9 1.43 0 0.606
10 -0.276 0 -0.319
# … with 190 more rowsGGally::ggpairs(df)Registered S3 method overwritten by 'GGally':
method from
+.gg ggplot2`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Employ MCCV: Predict Y from X, Y from Z, Y from X and Z
import pandas as pd
df = pd.DataFrame(data={'X' : X,'Y' : Y,'Z' : Z})
df.index.name = 'pt'
X = df.loc[:,['X']]
Y = df.loc[:,['Y']]
Z = df.loc[:,['Z']]
XZ = df.loc[:,['X','Z']]import mccv
mccv_YXobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YXobj.set_X(X)
mccv_YXobj.set_Y(Y)
mccv_YXobj.run_mccv()
mccv_YZobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YZobj.set_X(Z)
mccv_YZobj.set_Y(Y)
mccv_YZobj.run_mccv()
mccv_YXZobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YXZobj.set_X(XZ)
mccv_YXZobj.set_Y(Y)
mccv_YXZobj.run_mccv()
f_imp_dfs = dict()
f_imp_dfs['YXobj'] = \
mccv_YXobj.mccv_data['Feature Importance']
f_imp_dfs['YZobj'] = \
mccv_YZobj.mccv_data['Feature Importance']
f_imp_dfs['YXZobj'] = \
mccv_YXZobj.mccv_data['Feature Importance']Visualize feature importances
library(ggplot2)
f_imp_plot <- function(x,title){
ggplot(x,aes(feature,importance,color=feature)) +
geom_boxplot(alpha=0) +
geom_point(position=position_jitter(width=0.2)) +
scale_color_manual(values=c("X" = "indianred",
"Y" = "skyblue",
"Z" = "black",
"Intercept" = "gray")) +
theme_bw() +
labs(title=title)
}
library(patchwork)
(f_imp_plot( reticulate::py$f_imp_dfs$YXobj,"Y ~ X" ) +
f_imp_plot( reticulate::py$f_imp_dfs$YZobj,"Y ~ Z" ) ) /
f_imp_plot( reticulate::py$f_imp_dfs$YXZobj,"Y ~ X + Z" )Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
duplicated values: row names not set
Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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In this example, Z is confounding the relationship between X and Y. The ‘Y ~ X’ model shows that X is very important in predicting Y. However, when Z is included in the model ‘Y ~ X + Z’, you see that Z remains very important but X is no longer important for predicting Y. This toy example had Z causing Y and Z causing X, and any relationship between X and Y was spurious. Therefore, including X and Z as predictors showed only the importance of Z in predicting Y.
Generate dataset with an effect X, outcome Y, and a mediator Z variables
The random variable X generates Z which generates and Y. Any correlation between X and Y is therefore spurious
import numpy as np
N=100
X1 = np.random.normal(loc=0,scale=1,size=N)
X2 = np.random.normal(loc=1,scale=1,size=N)
X = np.concatenate([X1,X2])
import scipy as sc
Z = X + np.random.normal(loc=0,scale=1,size=len(X))
Y = np.concatenate([sc.stats.bernoulli.rvs(z,size=1) for z in (Z - min(Z)) / (max(Z) - min(Z))])df <- tibble::tibble(
X = reticulate::py$X,
Y = reticulate::py$Y,
Z = reticulate::py$Z
)
df[['Y']] <- factor(df$Y,levels=c(0,1))
df# A tibble: 200 × 3
X Y Z
<dbl[1d]> <fct> <dbl[1d]>
1 -0.566 1 -1.13
2 0.728 0 0.519
3 0.456 0 -0.657
4 -0.833 0 -0.908
5 -0.199 1 0.225
6 -0.241 0 -2.03
7 0.322 0 -0.649
8 -2.55 0 -2.90
9 0.350 0 -1.17
10 0.563 0 -0.117
# … with 190 more rowsGGally::ggpairs(df)Registered S3 method overwritten by 'GGally':
method from
+.gg ggplot2`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Employ MCCV: Predict Y from X, Y from Z, Y from X and Z
import pandas as pd
df = pd.DataFrame(data={'X' : X,'Y' : Y,'Z' : Z})
df.index.name = 'pt'
X = df.loc[:,['X']]
Y = df.loc[:,['Y']]
Z = df.loc[:,['Z']]
XZ = df.loc[:,['X','Z']]import mccv
mccv_YXobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YXobj.set_X(X)
mccv_YXobj.set_Y(Y)
mccv_YXobj.run_mccv()
mccv_YZobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YZobj.set_X(Z)
mccv_YZobj.set_Y(Y)
mccv_YZobj.run_mccv()
mccv_YXZobj = mccv.mccv(num_bootstraps=200,n_jobs=2)
mccv_YXZobj.set_X(XZ)
mccv_YXZobj.set_Y(Y)
mccv_YXZobj.run_mccv()
f_imp_dfs = dict()
f_imp_dfs['YXobj'] = \
mccv_YXobj.mccv_data['Feature Importance']
f_imp_dfs['YZobj'] = \
mccv_YZobj.mccv_data['Feature Importance']
f_imp_dfs['YXZobj'] = \
mccv_YXZobj.mccv_data['Feature Importance']Visualize feature importances
library(ggplot2)
f_imp_plot <- function(x,title){
ggplot(x,aes(feature,importance,color=feature)) +
geom_boxplot(alpha=0) +
geom_point(position=position_jitter(width=0.2)) +
scale_color_manual(values=c("X" = "indianred",
"Y" = "skyblue",
"Z" = "black",
"Intercept" = "gray")) +
theme_bw() +
labs(title=title)
}
library(patchwork)
(f_imp_plot( reticulate::py$f_imp_dfs$YXobj,"Y ~ X" ) +
f_imp_plot( reticulate::py$f_imp_dfs$YZobj,"Y ~ Z" ) ) /
f_imp_plot( reticulate::py$f_imp_dfs$YXZobj,"Y ~ X + Z" )Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
duplicated values: row names not set
Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
duplicated values: row names not set
Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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Warning in py_to_r.pandas.core.frame.DataFrame(object): index contains
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In this example, Z is a mediating relationship between X and Y. The ‘Y ~ X’ model shows that X is very important in predicting Y. However, when Z is included in the model ‘Y ~ X + Z’, you see that Z remains very important but X is no longer important for predicting Y. This toy example had Z causing Y and X relating to Y through Z, and any relationship between X and Y was mediated through Z. Therefore, including X and Z as predictors showed only the importance of Z in predicting Y.