% Rapport package team
% ANOVA Template
% 2011-04-26 20:25 CET
## 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.
-----------------------------------------------------
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
-----------------------------------------------------
Table: Table continues below
---------------------
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 0.168 3e-52
(Kolmogorov-Smirnov)
normality test
Anderson-Darling normality 18.75 7.261e-44
test
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 0.4629 0.4963
homogeneity of variances
Bartlett test of homogeneity 10.77 0.001032
of variances
--------------------------------------------------
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
[![](plots/ANOVA-5.png)](plots/ANOVA-5-hires.png)
### 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](http://en.wikipedia.org/wiki/Post-hoc_analysis) to explore patterns that were not specified a priori. Now we are presenting [Tukey's HSD test](http://en.wikipedia.org/wiki/Tukey%27s_range_test).
###### gender
----------------------------------------------------------
Difference Lower Bound Upper Bound
----------------- ------------ ------------- -------------
**female-male** -0.206 -0.543 0.132
----------------------------------------------------------
Table: Table continues below
---------------------------
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.
[![](plots/ANOVA-6.png)](plots/ANOVA-6-hires.png)
## 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.
------------------------------------------------------------
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
------------------------------------------------------------
Table: Table continues below
------------------------------------
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 0.168 3e-52
(Kolmogorov-Smirnov)
normality test
Anderson-Darling normality 18.75 7.261e-44
test
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 1.123 0.2892
homogeneity of variances
Bartlett test of homogeneity 11.13 0.0008509
of variances
--------------------------------------------------
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
[![](plots/ANOVA-7.png)](plots/ANOVA-7-hires.png)
### ANOVA Summary
#### ANOVA Table
------------------------------------------------------
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
------------------------------------------------------
Table: Table continues below
-----------------------------
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](http://en.wikipedia.org/wiki/Post-hoc_analysis) to explore patterns that were not specified a priori. Now we are presenting [Tukey's HSD test](http://en.wikipedia.org/wiki/Tukey%27s_range_test).
###### gender
----------------------------------------------------------
Difference Lower Bound Upper Bound
----------------- ------------ ------------- -------------
**female-male** -0.186 -0.538 0.165
----------------------------------------------------------
Table: Table continues below
---------------------------
P value
----------------- ---------
**female-male** _0.298_
---------------------------
There are no categories which differ significantly here.
###### partner
----------------------------------------------------------
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
----------------------------------------------------------
Table: Table continues below
-------------------------------------------------------
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
-------------------------------------------------------------
Difference Lower Bound
---------------------------------- ------------ -------------
**female:in a -0.014 -0.777
relationship-male:in a
relationship**
**male:married-male:in a -0.073 -1.25
relationship**
**female:married-male:in a -0.577 -1.877
relationship**
**male:single-male:in a 0.444 -0.23
relationship**
**female:single-male:in a 0.264 -0.545
relationship**
**male:married-female:in a -0.059 -1.266
relationship**
**female:married-female:in a -0.563 -1.89
relationship**
**male:single-female:in a 0.459 -0.267
relationship**
**female:single-female:in a 0.279 -0.574
relationship**
**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
-------------------------------------------------------------
Table: Table continues below
----------------------------------------------------------
Upper Bound P value
---------------------------------- ------------- ---------
**female:in a 0.749 _1_
relationship-male:in a
relationship**
**male:married-male:in a 1.103 _1_
relationship**
**female:married-male:in a 0.722 _0.801_
relationship**
**male:single-male:in a 1.119 _0.412_
relationship**
**female:single-male:in a 1.074 _0.938_
relationship**
**male:married-female:in a 1.148 _1_
relationship**
**female:married-female:in a 0.764 _0.83_
relationship**
**male:single-female:in a 1.184 _0.461_
relationship**
**female:single-female:in a 1.132 _0.938_
relationship**
**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.
[![](plots/ANOVA-8.png)](plots/ANOVA-8-hires.png)
-------
This report was generated with [R](http://www.r-project.org/) (3.0.1) and [rapport](https://rapporter.github.io/rapport/) (0.51) in _3.431_ sec on x86_64-unknown-linux-gnu platform.
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