Location Tests

Signed Rank Wilcoxon

Wilcoxon Signed Rank Test

Description

This is a nonparametric test for paired groups, i.e. does not assume normality in the data. Typically the t-Test is used for testing that the difference between population means for two paired samples are equal. But in cases where this assumption is violated, we can use the Wilcoxon Signed Rank Test. The test is based on analyzing the signs of the differences as well as the magnitude of the differences.

The test statistic is: $T = \sum_{i=1}^{n}sgn(X_{i})R|X_{i}|$ where $R|X_{i}|$ denotes the rank of $|X_{i}|,…,|X_{n}|$ where the rankings are from low to high.

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Used when assumptions for the t-Test are violated. However when normality assumptions are valid, the t-test is more powerful.

Tail	Alt. Hypothesis
$left\hspace{1cm}$	$H_{1}: \theta_{X} \lt \theta_{Y}:$ Distribution of $Y$ shifted to the right wrt to $X$
$right\hspace{1cm}$	$H_{1}: \theta_{X} \gt \theta_{Y}:$ Distribution of $Y$ shifted to the left wrt $X$
$both\hspace{1cm}$	$H_{1}: \theta_{X} \neq \theta_{Y}:$ Distribution of $Y$ shifted in either direction wrt $X$

Example

We consider the Charles Darwin’s Zea Mays dataset (easily available on the web).

Zea Mays Darwin

Select the region of data then go to Parietal -> LOC and set up a right-tailed test:

Signed Rank example 0a

Hit Probe to see the results without writing anything to the worksheet.

Signed Rank example 0b

Returns

p-value
decision
z-statistic

Reference: Hogg McKean Craig 8th

Sign Test

Description

The Sign Test is a nonparametric test which tests if the population means between two paired samples are equal. This is very much like the Wilcoxon Signed Rank Test but without the assumption of symmetric distribution of the differences around the median and without using the magnitude of the difference.

The sign statistic can be expressed as: $S(\theta_{0})=\sum_{i=1}^{n}I(X_{i}>\theta_{0})$

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Prefer Wilcoxon Signed Rank Test over Sign Test. Wilcoxon takes into account both the sign and magnitude of the difference for paired samples. The Sign Test only takes the difference of the sign into account.

Tail	Alt. Hypothesis
$left\hspace{1cm}$	$H_{1}: \theta_{X} \lt \theta_{Y}:$ Distribution of $Y$ shifted to the right wrt to $X$
$right\hspace{1cm}$	$H_{1}: \theta_{X} \gt \theta_{Y}:$ Distribution of $Y$ shifted to the left wrt $X$
$both\hspace{1cm}$	$H_{1}: \theta_{X} \neq \theta_{Y}:$ Distribution of $Y$ shifted in either direction wrt $X$

Example

We consider the Shoshoni rectangle dataset (easily available on the web) and the example 10.2.1 in Hogg McKean Craig 8th. The test the hypothesis: $H_{0}: \theta=0.618 \text{ versus } H_{1}: \theta \neq 0.618$ . We add a column called median to test against our hypothesized median of 0.618.

Zea Mays Darwin

Select the region of data then go to Parietal -> LOC and set up a two-tailed test:

Signed Rank example 0a

Hit Probe to see the results without writing anything to the worksheet.

Signed Rank example 0b

Returns

p-value
decision
z-statistic

Reference: Hogg McKean Craig 8th

T-Test

T Test

Description

Paired t-Test: Determines if there is a significant difference in the means two paired groups.

Assumptions:

Paired groups
No significant outliers
Differences are normally distributed

Tail	Alt. Hypothesis
$left\hspace{1cm}$	$H_{1}: \theta_{X} \lt \theta_{Y}:$ Population mean of $Y$ shifted to the right wrt to $X$
$right\hspace{1cm}$	$H_{1}: \theta_{X} \gt \theta_{Y}:$ Population mean of $Y$ shifted to the left wrt $X$
$both\hspace{1cm}$	$H_{1}: \theta_{X} \neq \theta_{Y}:$ Population mean of $Y$ shifted in either direction wrt $X$

Returns

p-value
decision
t-statistic

Rank Sum Test

Description

This is a nonparametric alternative to the two-sample t-Test. It is used to test if two groups are likely to derive from the same population. The measure for this is interpreted as comparing medians between the two groups.

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Also known as the Mann-Whitney U-test and Mann-Whitney-Wilcoxon test.

Tail	Alt. Hypothesis
$left\hspace{1cm}$	$H_{1}: \theta_{X} \lt \theta_{Y}:$ Median of $Y$ shifted to the right wrt to $X$
$right\hspace{1cm}$	$H_{1}: \theta_{X} \gt \theta_{Y}:$ Median of $Y$ shifted to the left wrt $X$
$both\hspace{1cm}$	$H_{1}: \theta_{X} \neq \theta_{Y}:$ Median of $Y$ shifted in either direction wrt $X$

Returns

p-value
decision
t-statistic

Z-Test

Z Test

Description

This one-sample location test determines if the mean of X derives from a normal distribution with mean $\mu$ and standard deviation $\sigma$ .

Tail	Alt. Hypothesis
$left\hspace{1cm}$	$H_{1}: \mu_{X} \lt \mu:$ Mean of $X$ shifted to the left wrt to $\mu$
$right\hspace{1cm}$	$H_{1}: \mu_{X} \gt \mu:$ Mean of $X$ shifted to the right wrt $\mu$
$both\hspace{1cm}$	$H_{1}: \mu_{X} \neq \mu:$ Mean of $X$ shifted in either direction wrt $\mu$

Returns

p-value
decision
z-statistic

Reference: Hogg McKean Craig 8th

Location Tests

Signed Rank Wilcoxon#

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Sign Test#

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T-Test#

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Rank Sum Test#

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Z-Test#

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Signed Rank Wilcoxon

Description

Example

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Sign Test

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Example

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T-Test

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Rank Sum Test

Description

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Z-Test

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