anova()
Description
Performs the analysis-of-variance (ANOVA) and analysis-of-covariance (ANCOVA).
Parameters
Input
anova(formula_like, data = {}, sum_of_squares = 3)
formula_like : A valid formula which will parse the data into a design matrix.
data : The dataframe which contains the data to be analyzed.
sum_of_squares : The type of sum of squares which is desired, the default is Type 3.
Returns
Returns an object with class “anova”; this object has accessible methods which are described below.
anova methods
results(return_type = “Dataframe”, decimals = 4, pretty_format = True)
return_type : The type of data structure the results should be returned as. Supported options are ‘Dataframe’ which will return a Pandas DataFrame or ‘Dictionary’ which will return a dictionary.
decimals : The number of decimal places the data should be rounded too.
pretty_format : If pretty formatting should be applied. This adds extra empty spaces in the returned data structure for visualization of the results.
regression_table(return_type = “Dataframe”, decimals = 4, conf_level = 0.95)
return_type : The type of data structure the results should be returned as. Supported options are ‘Dataframe’ which will return a Pandas DataFrame or ‘Dictionary’ which will return a dictionary.
decimals : The number of decimal places the data should be rounded too.
conf_level : The confidence interval desired.
predict(estimate = None)
estimate : Desired estimate. Available options are:
“y” or “xb” : Linear prediction
“residuals”, “res”, or “r” : Residuals
“standardized_residuals”, “standardized_r”, or “r_std” : Standardized residuals
“studentized_residuals”, “student_r”, or “r_stud” : Studentized (jackknifed) residuals
“leverage”, “lev” : Leverage of the observation (diagonal of the H matrix)
See predict() for formula information.
Effect Size Measures Formulas
By default, this method will return the measures of \(R^2\), \(\text{Adj. }R^2\), \(\eta^2\), \(\epsilon^2\), and \(\omega^2\). Please note that for the factor terms, the reported effect sizes are partial, i.e., \(\eta^2_p\), \(\epsilon^2_p\), and \(\omega^2_p\) respectively. See Olejnik and Aligna (2000) [1], Kelley and Preacher (2012) [2], and/or Grissom and Kim (2012) [3]
Additionally, \(R^2\) and \(\eta^2\) are the same but have different names due to coming from different frameworks which uses different terminology. Formulas for how to calculate these effect sizes comes from (Olejnik & Aligna, 2000) ; see
Eta-squared (\(\eta^2\)) and \(R^2\)
Adjusted \(R^2\)
Partial Eta-squared (\(\eta^2_p\))
Omega-squared (\(\omega^2\))
Partial Omega-squared (\(\omega^2_p\))
Where N is the total number of observations included in the model.
Examples
First to load required libraries for this example. Below, an example data set will be loaded in using statsmodels.datasets; the data loaded in is a data set available through Stata called ‘systolic’.
import researchpy as rp
import pandas as pd
# Used to load example data #
import statsmodels.datasets
systolic = statsmodels.datasets.webuse('systolic')
Now let’s get some quick information regarding the data set.
systolic.info()
<class 'pandas.core.frame.DataFrame'>
Int64Index: 58 entries, 0 to 57
Data columns (total 3 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 drug 58 non-null int16
1 disease 58 non-null int16
2 systolic 58 non-null int16
Now to take a look at the descriptive statistics of the univariate data. The output indicates that there are no missing observations and that each variable is stored as an integer.
rp.summarize(systolic["systolic"])
Name | N | Mean | Median | Variance | SD | SE | 95% Conf. Interval | |
---|---|---|---|---|---|---|---|---|
0 | systolic | 58 | 18.8793 | 21 | 163.862 | 12.8009 | 1.6808 | [15.5135, 22.2451] |
rp.crosstab(systolic["disease"], systolic["drug"])
Variable | Outcome | Count | Percent | |
---|---|---|---|---|
0 | drug | 4 | 16 | 27.59 |
1 | 2 | 15 | 25.86 | |
2 | 1 | 15 | 25.86 | |
3 | 3 | 12 | 20.69 | |
4 | disease | 3 | 20 | 34.48 |
5 | 2 | 19 | 32.76 | |
6 | 1 | 19 | 32.76 |
Now to conduct the ANOVA; by default Type 3 sum of squares are used. There are a few ways one can conduct an ANOVA using Researchpy, the suggested approach is to assign the ANOVA model to an object that way one can utilize the built-in methods. If one does not want to do that, then running the model with and displaying the results in one-line will work too; the output will be returned as a tuple. The suggested approach will be shown in this example.
m = anova("systolic ~ C(drug) + C(disease) + C(drug):C(disease)", data = systolic, sum_of_squares = 3)
desc, table = m.results()
print(desc, table, sep = "\n"*2)
Note: Effect size values for factors are partial.
Number of obs = | 58.0000 |
---|---|
Root MSE = | 10.5096 |
R-squared = | 0.4560 |
Adj R-squared = | 0.3259 |
Source | Sum of Squares | Degrees of Freedom | Mean Squares | F value | p-value | Eta squared | Omega squared |
---|---|---|---|---|---|---|---|
Model | 4,259.3385 | 11 | 387.2126 | 3.5057 | 0.0013 | 0.4560 | 0.3221 |
drug | 2,997.4719 | 3.0000 | 999.1573 | 9.0460 | 0.0001 | 0.3711 | 0.2939 |
disease | 415.8730 | 2.0000 | 207.9365 | 1.8826 | 0.1637 | 0.0757 | 0.0295 |
drug:disease | 707.2663 | 6.0000 | 117.8777 | 1.0672 | 0.3958 | 0.1222 | 0.0069 |
Residual | 5,080.8167 | 46 | 110.4525 | ||||
Total | 9,340.1552 | 57 | 163.8624 |
If it’s of interest, one can also access the underlying regression table.
m.regression_table()
systolic | Coef. | Std. Err. | t | p-value | 95% Conf. Interval |
---|---|---|---|---|---|
Intercept | 29.3333 | 4.2905 | 6.8367 | 0.0000 | [20.6969, 37.9697] |
drug | |||||
1 | (reference) | ||||
2 | -1.3333 | 6.3639 | -0.2095 | 0.8350 | [-14.1432, 11.4765] |
3 | -13.0000 | 7.4314 | -1.7493 | 0.0869 | [-27.9587, 1.9587] |
4 | -15.7333 | 6.3639 | -2.4723 | 0.0172 | [-28.5432, -2.9235] |
disease | |||||
1 | (reference) | ||||
2 | -1.0833 | 6.7839 | -0.1597 | 0.8738 | [-14.7387, 12.572] |
3 | -8.9333 | 6.3639 | -1.4038 | 0.1671 | [-21.7432, 3.8765] |
drug:disease | |||||
2:2 | 6.5833 | 9.7839 | 0.6729 | 0.5044 | [-13.1107, 26.2774] |
2:3 | -0.9000 | 8.9999 | -0.1000 | 0.9208 | [-19.0159, 17.2159] |
3:2 | -10.8500 | 10.2435 | -1.0592 | 0.2950 | [-31.4692, 9.7692] |
3:3 | 1.1000 | 10.2435 | 0.1074 | 0.9150 | [-19.5192, 21.7192] |
4:2 | 0.3167 | 9.3017 | 0.0340 | 0.9730 | [-18.4066, 19.04] |
4:3 | 9.5333 | 9.2022 | 1.0360 | 0.3056 | [-8.9897, 28.0564] |