#+TITLE: Rapport package team
#+AUTHOR: Normality Tests
#+DATE: 2011-04-26 20:25 CET
** Description
Overview of several normality tests and diagnostic plots that can screen
departures from normality.
*** Introduction
In statistics, /normality/ refers to an assumption that the distribution
of a random variable follows /normal/ (/Gaussian/) distribution. Because
of its bell-like shape, it's also known as the /"bell curve"/. The
formula for /normal distribution/ is:
$$f(x) = \frac{1}{\sqrt{2\pi{}\sigma{}^2}} e^{-\frac{(x-\mu{})^2}{2\sigma{}^2}}$$
/Normal distribution/ belongs to a /location-scale family/ of
distributions, as it's defined two parameters:
- $\mu$ - /mean/ or /expectation/ (location parameter)
- $\sigma^2$ - /variance/ (scale parameter)
[[plots/NormalityTest-1-hires.png][[[plots/NormalityTest-1.png]]]]
*** Normality Tests
**** Overview
Various hypothesis tests can be applied in order to test if the
distribution of given random variable violates normality assumption.
These procedures test the H_{0} that provided variable's distribution is
/normal/. At this point only few such tests will be covered: the ones
that are available in =stats= package (which comes bundled with default
R installation) and =nortest= package that is
[[http://cran.r-project.org/web/packages/nortest/index.html][available]]
on CRAN.
- *Shapiro-Wilk test* is a powerful normality test appropriate for
small samples. In R, it's implemented in =shapiro.test= function
available in =stats= package.
- *Lilliefors test* is a modification of /Kolmogorov-Smirnov test/
appropriate for testing normality when parameters or normal
distribution ($\mu$, $\sigma^2$) are not known. =lillie.test=
function is located in =nortest= package.
- *Anderson-Darling test* is one of the most powerful normality tests
as it will detect the most of departures from normality. You can find
=ad.test= function in =nortest= package.
**** Results
Here you can see the results of applied normality tests (/p-values/ less
than 0.05 indicate significant discrepancies):
We will use /Shapiro-Wilk/, /Lilliefors/ and /Anderson-Darling/ tests to
screen departures from normality in the response variable.
| 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.
*** Diagnostic Plots
There are various plots that can help you decide about the normality of
the distribution. Only a few most commonly used plots will be shown:
/histogram/, /Q-Q plot/ and /kernel density plot/.
**** Histogram
/Histogram/ was first introduced by /Karl Pearson/ and it's probably the
most popular plot for depicting the probability distribution of a random
variable. However, the decision depends on number of bins, so it can
sometimes be misleading. If the variable distribution is normal, bins
should resemble the "bell-like" shape.
[[plots/NormalityTest-2-hires.png][[[plots/NormalityTest-2.png]]]]
**** Q-Q Plot
"Q" in /Q-Q plot/ stands for /quantile/, as this plot compares empirical
and theoretical distribution (in this case, /normal/ distribution) by
plotting their quantiles against each other. For normal distribution,
plotted dots should approximate a "straight", =x = y= line.
[[plots/NormalityTest-3-hires.png][[[plots/NormalityTest-3.png]]]]
**** Kernel Density Plot
/Kernel density plot/ is a plot of smoothed /empirical distribution
function/. As such, it provides good insight about the shape of the
distribution. For normal distributions, it should resemble the well
known "bell shape".
[[plots/NormalityTest-4-hires.png][[[plots/NormalityTest-4.png]]]]
** Description
Overview of several normality tests and diagnostic plots that can screen
departures from normality.
*** Introduction
In statistics, /normality/ refers to an assumption that the distribution
of a random variable follows /normal/ (/Gaussian/) distribution. Because
of its bell-like shape, it's also known as the /"bell curve"/. The
formula for /normal distribution/ is:
$$f(x) = \frac{1}{\sqrt{2\pi{}\sigma{}^2}} e^{-\frac{(x-\mu{})^2}{2\sigma{}^2}}$$
/Normal distribution/ belongs to a /location-scale family/ of
distributions, as it's defined two parameters:
- $\mu$ - /mean/ or /expectation/ (location parameter)
- $\sigma^2$ - /variance/ (scale parameter)
*** Normality Tests
**** Overview
Various hypothesis tests can be applied in order to test if the
distribution of given random variable violates normality assumption.
These procedures test the H_{0} that provided variable's distribution is
/normal/. At this point only few such tests will be covered: the ones
that are available in =stats= package (which comes bundled with default
R installation) and =nortest= package that is
[[http://cran.r-project.org/web/packages/nortest/index.html][available]]
on CRAN.
- *Shapiro-Wilk test* is a powerful normality test appropriate for
small samples. In R, it's implemented in =shapiro.test= function
available in =stats= package.
- *Lilliefors test* is a modification of /Kolmogorov-Smirnov test/
appropriate for testing normality when parameters or normal
distribution ($\mu$, $\sigma^2$) are not known. =lillie.test=
function is located in =nortest= package.
- *Anderson-Darling test* is one of the most powerful normality tests
as it will detect the most of departures from normality. You can find
=ad.test= function in =nortest= package.
**** Results
Here you can see the results of applied normality tests (/p-values/ less
than 0.05 indicate significant discrepancies):
We will use /Shapiro-Wilk/, /Lilliefors/ and /Anderson-Darling/ tests to
screen departures from normality in the response variable.
| 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.
*** Diagnostic Plots
There are various plots that can help you decide about the normality of
the distribution. Only a few most commonly used plots will be shown:
/histogram/, /Q-Q plot/ and /kernel density plot/.
**** Histogram
/Histogram/ was first introduced by /Karl Pearson/ and it's probably the
most popular plot for depicting the probability distribution of a random
variable. However, the decision depends on number of bins, so it can
sometimes be misleading. If the variable distribution is normal, bins
should resemble the "bell-like" shape.
[[plots/NormalityTest-2-hires.png][[[plots/NormalityTest-2.png]]]]
**** Q-Q Plot
"Q" in /Q-Q plot/ stands for /quantile/, as this plot compares empirical
and theoretical distribution (in this case, /normal/ distribution) by
plotting their quantiles against each other. For normal distribution,
plotted dots should approximate a "straight", =x = y= line.
[[plots/NormalityTest-5-hires.png][[[plots/NormalityTest-5.png]]]]
**** Kernel Density Plot
/Kernel density plot/ is a plot of smoothed /empirical distribution
function/. As such, it provides good insight about the shape of the
distribution. For normal distributions, it should resemble the well
known "bell shape".
[[plots/NormalityTest-4-hires.png][[[plots/NormalityTest-4.png]]]]
** Description
Overview of several normality tests and diagnostic plots that can screen
departures from normality.
*** Introduction
In statistics, /normality/ refers to an assumption that the distribution
of a random variable follows /normal/ (/Gaussian/) distribution. Because
of its bell-like shape, it's also known as the /"bell curve"/. The
formula for /normal distribution/ is:
$$f(x) = \frac{1}{\sqrt{2\pi{}\sigma{}^2}} e^{-\frac{(x-\mu{})^2}{2\sigma{}^2}}$$
/Normal distribution/ belongs to a /location-scale family/ of
distributions, as it's defined two parameters:
- $\mu$ - /mean/ or /expectation/ (location parameter)
- $\sigma^2$ - /variance/ (scale parameter)
[[plots/NormalityTest-1-hires.png][[[plots/NormalityTest-1.png]]]]
*** Normality Tests
**** Overview
Various hypothesis tests can be applied in order to test if the
distribution of given random variable violates normality assumption.
These procedures test the H_{0} that provided variable's distribution is
/normal/. At this point only few such tests will be covered: the ones
that are available in =stats= package (which comes bundled with default
R installation) and =nortest= package that is
[[http://cran.r-project.org/web/packages/nortest/index.html][available]]
on CRAN.
- *Shapiro-Wilk test* is a powerful normality test appropriate for
small samples. In R, it's implemented in =shapiro.test= function
available in =stats= package.
- *Lilliefors test* is a modification of /Kolmogorov-Smirnov test/
appropriate for testing normality when parameters or normal
distribution ($\mu$, $\sigma^2$) are not known. =lillie.test=
function is located in =nortest= package.
- *Anderson-Darling test* is one of the most powerful normality tests
as it will detect the most of departures from normality. You can find
=ad.test= function in =nortest= package.
**** Results
Here you can see the results of applied normality tests (/p-values/ less
than 0.05 indicate significant discrepancies):
We will use /Shapiro-Wilk/, /Lilliefors/ and /Anderson-Darling/ tests to
screen departures from normality in the response variable.
| 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.
*** Diagnostic Plots
There are various plots that can help you decide about the normality of
the distribution. Only a few most commonly used plots will be shown:
/histogram/, /Q-Q plot/ and /kernel density plot/.
**** Histogram
/Histogram/ was first introduced by /Karl Pearson/ and it's probably the
most popular plot for depicting the probability distribution of a random
variable. However, the decision depends on number of bins, so it can
sometimes be misleading. If the variable distribution is normal, bins
should resemble the "bell-like" shape.
[[plots/NormalityTest-2-hires.png][[[plots/NormalityTest-2.png]]]]
**** Q-Q Plot
"Q" in /Q-Q plot/ stands for /quantile/, as this plot compares empirical
and theoretical distribution (in this case, /normal/ distribution) by
plotting their quantiles against each other. For normal distribution,
plotted dots should approximate a "straight", =x = y= line.
[[plots/NormalityTest-6-hires.png][[[plots/NormalityTest-6.png]]]]
**** Kernel Density Plot
/Kernel density plot/ is a plot of smoothed /empirical distribution
function/. As such, it provides good insight about the shape of the
distribution. For normal distributions, it should resemble the well
known "bell shape".
[[plots/NormalityTest-4-hires.png][[[plots/NormalityTest-4.png]]]]
--------------
This report was generated with [[http://www.r-project.org/][R]] (3.0.1)
and [[https://rapporter.github.io/rapport/][rapport]] (0.51) in /2.401/ sec on
x86\_64-unknown-linux-gnu platform.
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