ols()

Description

Conducts linear regression using the ordinary least squares approach.

Parameters

Input

ols(formula_like, data = {})

  • formula_like : A valid formula which will parse the data into a design matrix.

  • data : The dataframe which contains the data to be analyzed.

Returns

Returns an object with class “ols”; this object has accessible methods which are described below.

ols methods

  • results(return_type = “Dataframe”, decimals = 4, pretty_format = True, 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.

    • pretty_format : If pretty formatting should be applied. This adds extra empty spaces in the returned data structure for visualization of the results.

    • conf_level : The confidence interval desired.

-results- will return 3 objects, (1) is summary information, (2) is model table, and (3) is the regression table.

  • 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

Eta-squared (\(\eta^2\)) and \(R^2\)

\[\eta^2 = \frac{\text{SS}_{model}}{\text{SS}_{total}}\]

Adjusted \(R^2\)

\[\text{Adj. }R^2 = 1 - \frac{\text{df}_{total}}{\text{df}_{error}} * \frac{\text{SS}_{error}}{\text{SS}_{total}}\]

Omega-squared (\(\omega^2\))

\[\omega^2 = \frac{\text{SS}_{effect} - (\text{df}_{effect} * \text{MS}_{error})}{\text{SS}_{total} + \text{MS}_{error}}\]

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 fit the linear regression model, below is sample syntax.

m = ols("systolic ~ C(drug) + C(disease) + C(drug):C(disease)", data = systolic)

 desc, mod, table = m.results()
 print(desc, mod, table, sep = "\n"*2)
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 4259.3385 11 387.2126 3.5057 0.0013 0.456 0.3221
Residual 5080.8167 46 110.4525
Total 9340.1552 57 163.8624


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]

References

1

Stephen Olejnik and James Algina. Measures of effect size for comparative studies: applications, interpretations, and limitations. Contemporary Educational Psycholoty, 25:241–286, 2000. 10.1006/ceps.2000.1040.

2

Ken Kelly and Kristopher Preacher. On effect size. Psychological Methods, 7(2):137–152, 2012. 10.1006/ceps.2000.1040.

3

R. J. Grissom and J. J. Kim. Effect Sizes for Research: Univariate and Multivariate Applications. Routledge, second edition, 2012. ISBN 978-0-415-87769-5.