Commit 09b58ac6 authored by François  LAURENT's avatar François LAURENT
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StatsModels TP

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%% Cell type:markdown id:5162d7b2-625a-4699-922e-92c5c2bfa769 tags:
We will merely review statistical tests:
* Student $t$ tests
* compare a sample against the population mean
* compare two independent samples
* compare paired samples
* compare a sample mean against the population mean
* compare means of two independent samples
* compare the means of paired samples
* analyses of variance (one-way)
* compare more than two groups
* compare more than two group means
* tests for other tests' assumptions
* normality tests
* homoscedasticity tests
* $\chi^2$ tests for discrete variables
* goodness-of-fit test
* homogeneity and independence tests
* correlation coefficients
* correlation coefficient and linear regression
* effect sizes and test power
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## What Python can do -- What Python cannot
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| **sample** | a finite set of selected individuals<br />assumed to be representative of a population |
Always good to get a reminder about [general considerations](, __prior to data collection__ and analysis.
* Sampling from the population,
* identifying the sources of variability, etc.
* identifying the sources of variability,
* checking the assumptions of a test are met,
* etc.
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### Workflow
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To determine whether there is *sufficient evidence* to conclude the treatment has an effect, we use *statistical tests*.
However, because experimental designs are often complex and involve multiple treatments and additional sources of variability, most studies also involve multiple tests, that are usually carried out after a so-called *omnibus* test.
In addition, every statistical test makes various assumptions that in turn needs to be checked. As a consequence, every statistical analysis involves a series of tests and procedures.
In addition, every statistical test makes various assumptions that in turn needs to be checked.
As a consequence, reaching a conclusion about the data usually involves a series of tests and procedures.
<table style="text-align:left;"><tr><th>
Example worflow adapted from...?
<img src="img/example_anova_workflow.png" width="70%" />
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posthoc [label="Post-hoc tests"];
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### Replicability
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Not covered: tools and resources to support code and data management practices, some of which are provided by the Python ecosystem.
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## Data exploration
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``` python
print(f'{sample_mean:.2f} ± {1.96 * sem:.2f} years old on average')
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`scipy` actually offers a more straightforward way to computing confidence intervals:
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``` python
stats.norm(sample_mean, sem).interval(1 - alpha)
%%%% Output: execute_result
(45.535126163333835, 47.43629540529361)
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### Outliers
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``` python
import numpy as np
import pandas as pd
from matplotlib import pyplot as plt
import seaborn as sns
from scipy import stats
from patsy import dmatrices
import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.stats import diagnostic
from statsmodels.stats.multitest import multipletests
from statsmodels.stats.outliers_influence import OLSInfluence
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# Multi-way ANOVA
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## Q
Load the `titanic.csv` data file, insert the natural logarithm of `1+Fare` as a new column in the dataframe (*e.g.* with column name `'LogFare'`), and plot this new variable as a function of `Age`, `Pclass` and `Sex`.
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## A
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``` python
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## Q
Fit a linear model to these data to explain our synthetic variable `LogFare` as a function of `Age`, `Pclass` and `Sex`.
Treat `Pclass` and `Sex` as factors.
Print an ANOVA table.
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## A
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``` python
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## Q
Let us ignore the not-normal residuals and play with post-hoc tests instead.
Split the ANOVA for levels of `Pclass` and `Sex`, perform all pairwise comparisons if it make sense, and correct for multiple comparisons.
We are not interested in the significance of the slope of `Age` for the different levels of the factors.
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## A
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``` python
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# Linear model with multiple variables
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## Q
Load the `mi.csv` file and plot the variables `Temperature`, `HeartRate` and `PhysicalActivity`.
We will try to «explain» `Temperature` from `HeartRate` and `PhysicalActivity`.
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## A
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``` python
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## Q
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The `PhysicalActivity` variable exhibit a long-tail distribution. This is usually undesirable for an explanatory variable, because we cannot densely sample a large part of its domain of possible values, and therefore a model based on the data cannot be reliable.
We will proceed to transforming `PhysicalActivity` using a simple natural logarithm. `log` is undefined at $0$ and tends to the infinite near $0$, which renders its straightforward application to `PhysicalActivity` inappropriate. Therefore we will also add $1$ to the `PhysicalActivity` measurements prior to applying `log`.
Plot again the temperature versus the transformed `PhysicalActivity` variable and compare the skewness of the transformed versus raw variable.
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## A
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``` python
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## Q
To appreciate the increased robustness of a linear model using the transformed variable compared to the raw variable, design a simple univariate linear regression of `Temperature` as response variable, and draw the Cook's distance of all the observations in regard of this model:
* first with the raw `PhysicalActivity` as explanatory variable,
* second with the transformed `PhysicalActivity` as explanatory variable.
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## A
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``` python
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## Q
Make a linear model of `Temperature` as response and `HeartRate` and `PhysicalActivity` (or its transformed variant) as explanatory variables.
Make two such models, one with interaction and one without. How would you choose between the two models?
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## A (with nested Q&A)
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``` python
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``` python
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### Q
To get a better intuition about the log-likelihood, plot it (with a dot plot) for different models, with one variable, with two variables, with and without interaction.
Feel free to introduce one or two extra explanatory variables such as `BMI`.
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### A
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``` python
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# White test for homoscedasticity
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To keep things simple, let us use the `'Heart + PhysicalActivity'` or `'Heart + logPhysicalActivity'`.
## Q
Inspect the residuals plotting them versus each explanatory variable.
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## A
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``` python
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## Q
We will further inspect the residuals for heteroscedasticity, using the [White test](
`statsmodels` features an implementation of this test, but the [documentation]( is scarce on details.
Try to apply the `het_white` function, but do not feel ashamed if you fail.
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## A
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``` python
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## Q
Instead, we will implement this test, as an application of polynomial regression.
The algorithm is simple. First part:
* take the squared residuals as a response variable,
* take the same explanatory variables as in the original model, plus all their possible interaction terms, plus all their values squared,
* fit a linear model to these data.
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## A
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``` python
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## Q
Second part:
* get the coefficient of determination $R^2$,
* get the sample size $n$,
* set the number $k$ of degrees of freedom as the number of predictors (intercept excluded),
The test is:
H_0: nR^2 \sim \chi_{k}^2
H_A: nR^2 > \tt{Critical Value}(\chi_{k}^2, 1-\alpha)
You do not necessarily need to compute the critical value. Just note the test is one-sided.
Compute the statistic $nR^2$ and the resulting $p$-value.
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## A
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``` python
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