Description

An ANOVA report with table of descriptives, diagnostic tests and ANOVA-specific statistics.

Introduction

Analysis of Variance or ANOVA is a statistical procedure that tests equality of means for several samples. It was first introduced in 1921 by famous English statistician Sir Ronald Aylmer Fisher.

Model Overview

One-Way ANOVA was carried out, with Gender as independent variable, and Internet usage in leisure time (hours per day) as a response variable. Factor interaction was taken into account.

Descriptives

In order to get more insight on the model data, a table of frequencies for ANOVA factors is displayed, as well as a table of descriptives.

Frequency Table

Below lies a frequency table for factors in ANOVA model. Note that the missing values are removed from the summary.

gender N % Cumul. N Cumul. %
male 410 60.92 410 60.92
female 263 39.08 673 100
Total 673 100 673 100

Descriptive Statistics

The following table displays the descriptive statistics of ANOVA model. Factor levels lie on the left-hand side, while the corresponding statistics for response variable are given on the right-hand side.

Table continues below
Gender Min Max Mean Std.Dev. Median IQR
male 0 12 3.27 1.953 3 3
female 0 12 3.064 2.355 2 3
Skewness Kurtosis
0.9443 0.9858
1.398 1.87

Diagnostics

Before we carry out ANOVA, we'd like to check some basic assumptions. For those purposes, normality and homoscedascity tests are carried out alongside several graphs that may help you with your decision on model's main assumptions.

Diagnostics

Univariate Normality
Method Statistic p-value
Lilliefors (Kolmogorov-Smirnov) normality test 0.168 3e-52
Anderson-Darling normality test 18.75 7.261e-44
Shapiro-Wilk normality test 0.9001 1.618e-20

So, the conclusions we can draw with the help of test statistics:

• based on Lilliefors test, distribution of Internet usage in leisure time (hours per day) is not normal

• Anderson-Darling test confirms violation of normality assumption

• according to Shapiro-Wilk test, the distribution of Internet usage in leisure time (hours per day) is not normal

As you can see, the applied tests confirm departures from normality of the Internet usage in leisure time (hours per day).

Homoscedascity

In order to test homoscedascity, Bartlett and Fligner-Kileen tests are applied.

Method Statistic p-value
Fligner-Killeen test of homogeneity of variances 0.4629 0.4963
Bartlett test of homogeneity of variances 10.77 0.001032

When it comes to equality of variances, applied tests yield inconsistent results. While Fligner-Kileen test confirmed the hypotheses of homoscedascity, Bartlett's test rejected it.

Diagnostic Plots

Here you can see several diagnostic plots for ANOVA model:

• residuals against fitted values
• scale-location plot of square root of residuals against fitted values
• normal Q-Q plot
• residuals against leverages

ANOVA Summary

ANOVA Table

Df Sum.Sq Mean.Sq F.value Pr..F.
gender 1 6.422 6.422 1.43 0.2322
Residuals 636 2856 4.49

F-test for Gender is not statistically significant, which implies that there is no Gender effect on response variable.

Post Hoc test

Results

After getting the results of the ANOVA, usually it is advisable to run a post hoc test to explore patterns that were not specified a priori. Now we are presenting Tukey's HSD test.

gender
Table continues below
Difference Lower Bound Upper Bound
female-male -0.206 -0.543 0.132
P value
female-male 0.232

There are no categories which differ significantly here.

Plot

Below you can see the result of the post hoc test on a plot.

Description

An ANOVA report with table of descriptives, diagnostic tests and ANOVA-specific statistics.

Introduction

Analysis of Variance or ANOVA is a statistical procedure that tests equality of means for several samples. It was first introduced in 1921 by famous English statistician Sir Ronald Aylmer Fisher.

Model Overview

Two-Way ANOVA was carried out, with Gender and Relationship status as independent variables, and Internet usage in leisure time (hours per day) as a response variable. Factor interaction was taken into account.

Descriptives

In order to get more insight on the model data, a table of frequencies for ANOVA factors is displayed, as well as a table of descriptives.

Frequency Table

Below lies a frequency table for factors in ANOVA model. Note that the missing values are removed from the summary.

gender partner N % Cumul. N Cumul. %
male in a relationship 150 23.7 150 23.7
female in a relationship 120 18.96 270 42.65
male married 33 5.213 303 47.87
female married 29 4.581 332 52.45
male single 204 32.23 536 84.68
female single 97 15.32 633 100
Total Total 633 100 633 100

Descriptive Statistics

The following table displays the descriptive statistics of ANOVA model. Factor levels and their combinations lie on the left-hand side, while the corresponding statistics for response variable are given on the right-hand side.

Table continues below
Gender Relationship status Min Max Mean Std.Dev.
male in a relationship 0.5 12 3.058 1.969
male married 0 8 2.985 2.029
male single 0 10 3.503 1.936
female in a relationship 0.5 10 3.044 2.216
female married 0 10 2.481 1.967
female single 0 12 3.323 2.679
Median IQR Skewness Kurtosis
2.5 2 1.324 2.649
3 2 0.862 0.1509
3 3 0.7574 0.08749
3 3 1.383 1.831
2 1.75 2.063 5.586
3 3.5 1.185 0.9281

Diagnostics

Before we carry out ANOVA, we'd like to check some basic assumptions. For those purposes, normality and homoscedascity tests are carried out alongside several graphs that may help you with your decision on model's main assumptions.

Diagnostics

Univariate Normality
Method Statistic p-value
Lilliefors (Kolmogorov-Smirnov) normality test 0.168 3e-52
Anderson-Darling normality test 18.75 7.261e-44
Shapiro-Wilk normality test 0.9001 1.618e-20

So, the conclusions we can draw with the help of test statistics:

• based on Lilliefors test, distribution of Internet usage in leisure time (hours per day) is not normal

• Anderson-Darling test confirms violation of normality assumption

• according to Shapiro-Wilk test, the distribution of Internet usage in leisure time (hours per day) is not normal

As you can see, the applied tests confirm departures from normality of the Internet usage in leisure time (hours per day).

Homoscedascity

In order to test homoscedascity, Bartlett and Fligner-Kileen tests are applied.

Method Statistic p-value
Fligner-Killeen test of homogeneity of variances 1.123 0.2892
Bartlett test of homogeneity of variances 11.13 0.0008509

When it comes to equality of variances, applied tests yield inconsistent results. While Fligner-Kileen test confirmed the hypotheses of homoscedascity, Bartlett's test rejected it.

Diagnostic Plots

Here you can see several diagnostic plots for ANOVA model:

• residuals against fitted values
• scale-location plot of square root of residuals against fitted values
• normal Q-Q plot
• residuals against leverages

ANOVA Summary

ANOVA Table

Table continues below
Df Sum.Sq Mean.Sq F.value
gender 1 4.947 4.947 1.085
partner 2 31.21 15.61 3.424
gender:partner 2 3.038 1.519 0.3332
Residuals 593 2703 4.558
Pr..F.
gender 0.2979
partner 0.03324
gender:partner 0.7168
Residuals

F-test for Gender is not statistically significant, which implies that there is no Gender effect on response variable. Effect of Relationship status on response variable is significant. Interaction between levels of Gender and Relationship status wasn't found significant (p = 0.717).

Post Hoc test

Results

After getting the results of the ANOVA, usually it is advisable to run a post hoc test to explore patterns that were not specified a priori. Now we are presenting Tukey's HSD test.

gender
Table continues below
Difference Lower Bound Upper Bound
female-male -0.186 -0.538 0.165
P value
female-male 0.298

There are no categories which differ significantly here.

partner
Table continues below
Difference Lower Bound
married-in a relationship -0.289 -1.012
single-in a relationship 0.371 -0.061
single-married 0.66 -0.059
Upper Bound P value
married-in a relationship 0.435 0.616
single-in a relationship 0.803 0.109
single-married 1.379 0.079

There are no categories which differ significantly here.

gender:partner
Table continues below
Difference Lower Bound
female:in a relationship-male:in a relationship -0.014 -0.777
male:married-male:in a relationship -0.073 -1.25
female:married-male:in a relationship -0.577 -1.877
male:single-male:in a relationship 0.444 -0.23
female:single-male:in a relationship 0.264 -0.545
male:married-female:in a relationship -0.059 -1.266
female:married-female:in a relationship -0.563 -1.89
male:single-female:in a relationship 0.459 -0.267
female:single-female:in a relationship 0.279 -0.574
female:married-male:married -0.504 -2.105
male:single-male:married 0.518 -0.635
female:single-male:married 0.338 -0.899
male:single-female:married 1.022 -0.256
female:single-female:married 0.842 -0.512
female:single-male:single -0.18 -0.955
Upper Bound P value
female:in a relationship-male:in a relationship 0.749 1
male:married-male:in a relationship 1.103 1
female:married-male:in a relationship 0.722 0.801
male:single-male:in a relationship 1.119 0.412
female:single-male:in a relationship 1.074 0.938
male:married-female:in a relationship 1.148 1
female:married-female:in a relationship 0.764 0.83
male:single-female:in a relationship 1.184 0.461
female:single-female:in a relationship 1.132 0.938
female:married-male:married 1.097 0.946
male:single-male:married 1.67 0.794
female:single-male:married 1.575 0.971
male:single-female:married 2.3 0.201
female:single-female:married 2.196 0.481
female:single-male:single 0.594 0.986

There are no categories which differ significantly here.

Plot

Below you can see the result of the post hoc test on a plot.

This report was generated with R (3.0.1) and rapport (0.51) in 3.431 sec on x86_64-unknown-linux-gnu platform.

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