Lab 6

Lab 6 October 3, 2022 1 Lab 6 In this lab we discuss statistical modelling and least squares regression. 1.1 Simple Linear Regression [1]: import numpy as np import pandas as pd import seaborn as sns import matplotlib.pyplot as plt from scipy import stats [2]: # Read .csv data possum = pd . read_csv( "possum.csv" ) # The possum dataset consists of morphometric measurements on 46 possums. possum . head() [2]: sex age headL skullW totalL tailL 0 m 8 94.1 60.4 89.0 36.0 1 f 6 92.5 57.6 91.5 36.5 2 f 6 94.0 60.0 95.5 39.0 3 f 6 93.2 57.1 92.0 38.0 4 f 2 91.5 56.3 85.5 36.0 stats.linregress: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.linregress.html [3]: # Building a simple model with only one term # Simple linear regression is a linear regression model with a single ␣ , → explanatory variable. model = stats . linregress(x = possum[ ' age ' ], y = possum[ ' headL ' ]) [4]: # Finding the slope of the regression line model . slope [4]: 0.5631158455392805 [5]: # Finding the intercept of the regression line model . intercept [5]: 90.08288948069242 1

[6]: # Finding the correlation coefficient (R) model . rvalue [6]: 0.4011016610119052 [7]: # Calculating the R-squared (model . rvalue ** 2 ) [7]: 0.16088254246650932 [8]: # Plotting the fitted line X = pd . DataFrame({ "age" :np . linspace( 0 , 9 , 45 )}) y_pred = model . intercept + model . slope * X plt . scatter(possum[ ' age ' ], possum[ ' headL ' ], label = ' original data ' ) plt . plot(X, y_pred, color = ' red ' , label = ' fitted line ' ) plt . xlabel( "age" ) plt . ylabel( "headL" ) plt . legend() plt . show() sns.lmplot: https://seaborn.pydata.org/generated/seaborn.lmplot.html [9]: # Alternative way to plot the fitted line sns . lmplot(x = "age" , y = "headL" , data = possum, ci = None ) plt . xlim( 0 , 10 ) plt . show() 2

1.2 Effect of an Influential Point [11]: # Let ' s look at the last 5 rows of the original dataset! # The original dataset has 46 rows and 6 columns. possum . tail() [11]: sex age headL skullW totalL tailL 41 m 3 85.3 54.1 77.0 32.0 42 f 2 90.0 55.5 81.0 32.0 43 m 3 85.1 51.5 76.0 35.5 44 m 3 90.7 55.9 81.0 34.0 45 m 2 91.4 54.4 84.0 35.0 pd.DataFrame.copy: https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.copy.html [12]: # Now we would like to add an influential point and see how this affects the ␣ , → regression line. # We decide to add a point that behaves like an outlier to the bottom of the ␣ , → dataframe. 4

# Let ' s add a point representing "age" = 9 and "headL" = 86. # We call the new dataframe "possum_modified". possum_modified = possum . copy() possum_modified . loc[ 46 , ' age ' ] = 9 possum_modified . loc[ 46 , ' headL ' ] = 86 [13]: # Now let ' s look at the last 5 rows of the modified dataframe! possum_modified . tail() [13]: sex age headL skullW totalL tailL 42 f 2.0 90.0 55.5 81.0 32.0 43 m 3.0 85.1 51.5 76.0 35.5 44 m 3.0 90.7 55.9 81.0 34.0 45 m 2.0 91.4 54.4 84.0 35.0 46 NaN 9.0 86.0 NaN NaN NaN [14]: # The new point that is added to the dataframe is identified in red in the ␣ , → scatterplot below. plt . scatter(possum_modified . loc[ 0 : 45 , ' age ' ], possum_modified . loc[ 0 : 45 , ␣ , → ' headL ' ], color = ' blue ' ) plt . scatter(possum_modified . loc[ 46 , ' age ' ], possum_modified . loc[ 46 , ' headL ' ], ␣ , → color = ' red ' ) plt . xlabel( "age" ) plt . ylabel( "headL" ) plt . show() 5

[21]: # previous fitted using original data sns . lmplot(x = "age" , y = "headL" , data = possum, ci = None ) plt . xlim( 0 , 10 ) plt . show() 7

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