Load Data<\/strong><\/p>\n The data we will use is the iris data set, available in the numpy<\/strong> library.<\/p>\n Exploratory Data Analysis<\/strong><\/p>\n The Jupyter Notebook I’ve made available on GITHUB<\/a><\/span><\/strong> has a number of EDA steps. For this tutorial the variable we will look at is Sepal Length, which appears to different between Species.<\/p>\n T-test<\/strong><\/p>\n We will start by conducting a t-test. Since a t-test is a comparison of means between two groups, I’ll create a data set with only the setosa and versicolor species.<\/p>\n First I build the t-test in two common stats libraries in python, statsmodels<\/strong> and scipy<\/strong>.<\/p>\n Unfortunately, the output of both of these approaches leaves a lot to be desired. They simply return the t-statistics, p-value, and degrees of freedom.<\/p>\n To get a better look at the underlying comparison, I’ll instead fit the t-test using the researchpy <\/strong>library.<\/p>\n This output is more informative. We can see the summary stats for both groups at the top. the t-test results follow below. We see the observed difference, versicolor has a sepal length 0.93 (5.006 – 5.936) longer that setosa, on average. We also get the degrees of freedom, t-statistic, and p-value, along with several measures of effect size.<\/p>\n Linear Regression<\/strong><\/p>\n Now that we see what the output looks like, let’s confirm that this is indeed just linear regression!<\/p>\n We fit our model using the statsmodels <\/strong>library.<\/p>\n Notice that the slope coefficient for versicolor is 0.93, indicating it’s sepal length is, on average, 0.93 greater than setosa’s sepal length. This is the same result we obtained with our t-test above.<\/p>\n The intercept coefficient is 5.006, which means that when versicolor is set to “0” in the model (0 * 0.93 = 0) all we are left with is the intercept, which is the mean value for setosa’s sepal length, the same as we saw in our t-test.<\/p>\n What about the residuals?<\/strong><\/p>\n The question original question was about residuals from the t-test. Recall, the residuals are the difference between actual\/observed value and the predicted value. When we have a simple linear regression with two levels (a t-test) the predicted value is simply the overall mean value for that group.<\/p>\n We can add predictions from the linear regression model into our data set and calculate the residuals, plot the residuals, and then calculate the mean squared error.<\/p>\n In the code, you will find the same approach taken by just applying the group mean as the “predicted” value, which is the same value that the model predicts. At the bottom of the code, we will find that the outcome of the MSE is the same.<\/p>\n![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2022\/05\/Screen-Shot-2022-05-15-at-5.17.06-PM-1024x386.png\")
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<\/a><\/p>\n\r\n## get two groups to compare\r\ntwo_groups = ["setosa", 'versicolor']\r\n\r\n## create a data frame of the two groups\r\ndf2 = df[df['species'].isin(two_groups)]\r\n<\/pre>\n
\r\n## t-test in statsmodels.api\r\nsmf.stats.ttest_ind(x1 = df2[df['species'] == 'setosa']['sepal_length'],\r\n x2 = df2[df['species'] == 'versicolor']['sepal_length'],\r\n alternative="two-sided")\r\n\r\n## t-test in scipy\r\nstats.ttest_ind(a = df2[df['species'] == 'setosa']['sepal_length'],\r\n b = df2[df['species'] == 'versicolor']['sepal_length'],\r\n alternative="two-sided")\r\n<\/pre>\n
\r\n## t-test in reserachpy\r\nrp.ttest(group1 = df2[df['species'] == 'setosa']['sepal_length'],\r\n group2 = df2[df['species'] == 'versicolor']['sepal_length'])\r\n<\/pre>\n
![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2022\/05\/Screen-Shot-2022-05-15-at-5.39.03-PM-1024x434.png\")
<\/a><\/p>\n\r\n## Linear model to compare results with t-test (convert the species types of dummy variables)\r\nX = df2[['species']]\r\nX = pd.get_dummies(X['species'], drop_first = True)\r\n\r\ny = df2[['sepal_length']]\r\n\r\n## add an intercept constant, since it isn't done automatically\r\nX = smf.add_constant(X)\r\n\r\n# Build regression model\r\nfit_lm = smf.OLS(y, X).fit()\r\n\r\n# Get model output\r\n\r\nfit_lm.summary()\r\n<\/pre>\n
![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2022\/05\/Screen-Shot-2022-05-15-at-5.43.18-PM-1024x874.png\")
<\/a><\/p>\n\r\n## Add the predictions back to the data set\r\ndf2['preds'] = fit_lm.predict(X)\r\n\r\n## Calculate the residual\r\nresid = df2['sepal_length'] - df2['preds']\r\n\r\n## plot the residuals\r\nsns.kdeplot(resid,shade = True)\r\nplt.axvline(x = 0,linewidth=4, linestyle = '--', color='r')\r\n<\/pre>\n
![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2022\/05\/Screen-Shot-2022-05-15-at-5.48.29-PM.png\")
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