This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
Sign test
Pearson correlation
$z$ test for the difference between two proportions
One categorical with $J$ independent groups ($J \geqslant 2$)
One quantitative of interval or ratio level
Null hypothesis
Null hypothesis
Null hypothesis
Null hypothesis
Null hypothesis
H_{0}: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
H_{0}: the population median of the difference scores is equal to zero
A difference score is the difference between the first score of a pair and the second score of a pair.
H_{0}: $\rho = \rho_0$
$\rho$ is the unknown Pearson correlation in the population, $\rho_0$ is the correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.
H_{0}: $\pi_1 = \pi_2$
$\pi_1$ is the population proportion of 'successes' for group 1; $\pi_2$ is the population proportion of 'successes' for group 2
H_{0}: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
or equivalently
H_{0}: the probability of drawing an observation from condition 1 is $\pi_1$, the probability of drawing an observation from condition 2 is $\pi_2$, $\ldots$,
the probability of drawing an observation from condition $J$ is $\pi_J$
H_{0}: $m = 0$
$m$ is the population median of the difference scores. A difference score is the difference between the first score of a pair and the second score of a pair.
Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
H_{1} two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair)
H_{1} right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair)
H_{1} left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
H_{1} two sided: the population median of the difference scores is different from zero
H_{1} right sided: the population median of the difference scores is larger than zero
H_{1} left sided: the population median of the difference scores is smaller than zero
H_{1} two sided: $\rho \neq \rho_0$
H_{1} right sided: $\rho > \rho_0$
H_{1} left sided: $\rho < \rho_0$
H_{1} two sided: $\pi_1 \neq \pi_2$
H_{1} right sided: $\pi_1 > \pi_2$
H_{1} left sided: $\pi_1 < \pi_2$
H_{1}: the population proportions are not all as specified under the null hypothesis
or equivalently
H_{1}: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
H_{1} two sided: $m \neq 0$
H_{1} right sided: $m > 0$
H_{1} left sided: $m < 0$
Assumptions
Assumptions of test for correlation
Assumptions
Assumptions
Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions)
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb:
Significance test: number of successes and number of failures are each 5 or more in both sample groups
Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups
Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more
Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
Sample size is large enough for $X^2$ to be approximately chisquared distributed. Rule of thumb: all $J$ expected cell counts are 5 or more
Sample is a simple random sample from the population. That is, observations are independent of one another
The population distribution of the difference scores is symmetric
Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
Note: sometimes it considered sufficient for the data to be measured at an ordinal scale, rather than an interval or ratio scale. However, since the test statistic is based on ranked difference scores, we need to know whether a change in scores from, say, 6 to 7 is larger than/smaller than/equal to a change from 5 to 6. This is impossible to know for ordinal scales, since for these scales the size of the difference between values is meaningless.
Test statistic
Test statistic
Test statistic
Test statistic
Test statistic
$W = $ number of difference scores that is larger than 0
Test statistic for testing H0: $\rho = 0$:
$t = \dfrac{r \times \sqrt{N  2}}{\sqrt{1  r^2}} $
where $r$ is the sample correlation $r = \frac{1}{N  1} \sum_{j}\Big(\frac{x_{j}  \bar{x}}{s_x} \Big) \Big(\frac{y_{j}  \bar{y}}{s_y} \Big)$ and $N$ is the sample size
Test statistic for testing values for $\rho$ other than $\rho = 0$:
$r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$, where $r$ is the sample correlation
$\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1  \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
$z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$,
$p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$,
$p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$,
$n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2
Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1$
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells
Two different types of test statistics can be used; both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
For each subject, compute the sign of the difference score $\mbox{sign}_d = \mbox{sgn}(\mbox{score}_2  \mbox{score}_1)$. The sign is 1 if the difference is larger than zero, 1 if the diffence is smaller than zero, and 0 if the difference is equal to zero.
For each subject, compute the absolute value of the difference score $\mbox{score}_2  \mbox{score}_1$.
Exclude subjects with a difference score of zero. This leaves us with a remaining number of difference scores equal to $N_r$.
Assign ranks $R_d$ to the $N_r$ remaining absolute difference scores. The smallest absolute difference score corresponds to a rank score of 1, and the largest absolute difference score corresponds to a rank score of $N_r$. If there are ties, assign them the average of the ranks they occupy.
Then compute the test statistic:
$W_1 = \sum\, R_d^{+}$
or
$W_1 = \sum\, R_d^{}$
That is, sum all ranks corresponding to a positive difference or sum all ranks corresponding to a negative difference. Theoratically, both definitions will result in the same test outcome. However:
tables with critical values for $W_1$ are usually based on the smaller of $\sum\, R_d^{+}$ and $\sum\, R_d^{}$. So if you are using such a table, pick the smaller one.
If you are using the normal approximation to find the $p$ value, it makes things most straightforward if you use $W_1 = \sum\, R_d^{+}$ (if you use $W_1 = \sum\, R_d^{}$, the right and left sided alternative hypotheses 'flip').
$W_2 = \sum\, \mbox{sign}_d \times R_d$
That is, for each remaining difference score, multiply the rank of the absolute difference score by the sign of the difference score, and then sum all of the products.
Sampling distribution of $W_1$ and of $W_2$ if H_{0} were true
The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $p$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $p = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $np = n \times 0.5$ and standard deviation $\sqrt{np(1p)} = \sqrt{n \times 0.5(1  0.5)}$. Hence, if $n$ is large, the standardized test statistic
$$z = \frac{W  n \times 0.5}{\sqrt{n \times 0.5(1  0.5)}}$$
follows approximately the standard normal distribution if the null hypothesis were true.
Sampling distribution of $t$:
$t$ distribution with $N  2$ degrees of freedom
Sampling distribution of $z$:
Approximately the standard normal distribution
Approximately the standard normal distribution
Approximately the chisquared distribution with $J  1$ degrees of freedom
Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here
$$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$
$$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$
Hence, if $N_r$ is large, the standardized test statistic
$$z = \frac{W_1  \mu_{W_1}}{\sigma_{W_1}}$$
follows approximately the standard normal distribution if the null hypothesis were true.
Sampling distribution of $W_2$:
If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here
$$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$
Hence, if $N_r$ is large, the standardized test statistic
$$z = \frac{W_2}{\sigma_{W_2}}$$
follows approximately the standard normal distribution if the null hypothesis were true.
If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used.
Note: the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated if ties are present in the data.
Significant?
Significant?
Significant?
Significant?
Significant?
If $n$ is small, the table for the binomial distribution should be used:
Two sided:
Check if $W$ observed in sample is in the rejection region or
Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Right sided:
Check if $W$ observed in sample is in the rejection region or
Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Left sided:
Check if $W$ observed in sample is in the rejection region or
Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
If $n$ is large, the table for standard normal probabilities can be used:
Two sided:
Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Right sided:
Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Left sided:
Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
$t$ Test two sided:
Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
$t$ Test right sided:
Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
$t$ Test left sided:
Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
$z$ Test two sided:
Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
$z$ Test right sided:
Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
$z$ Test left sided:
Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Two sided:
Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Right sided:
Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Left sided:
Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$ and the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval).
Then transform back to get the approximate $C$% confidence interval for $\rho$:
$(p_1  p_2) \pm z^* \times \sqrt{\dfrac{p_1(1  p_1)}{n_1} + \dfrac{p_2(1  p_2)}{n_2}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval)
With plus four method:
$(p_{1.plus}  p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1  p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1  p_{2.plus})}{n_2 + 2}}$
where $p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}$, $p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}$, and the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval)


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Properties of the Pearson correlation coefficient
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The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables.
The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable.
The Pearson correlation coefficient can take on values between 1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship.
The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example:
the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y  6$.
the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $3x + 5$ and $2y  6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $3$.
The Pearson correlation coefficient does not say anything about causality.
The Pearson correlation coefficient is sensitive to outliers.
Do people tend to score higher on mental health after a mindfulness course?
Is there a linear relationship between physical health and mental health?
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.
Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?
Is the median of the differences between the mental health scores before and after an intervention different from 0?
Put the two paired variables in the boxes below Variable 1 and Variable 2
Under Test Type, select the Sign test
Analyze > Correlate > Bivariate...
Put your two variables in the box below Variables
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s)
Click the Statistics... button, and click on the square in front of Chisquare
Put your categorical variable in the box below Test Variable List
Fill in the population proportions / probabilities according to $H_0$ in the box below Expected Values. If $H_0$ states that they are all equal, just pick 'All categories equal' (default)
Put the two paired variables in the boxes below Variable 1 and Variable 2
Under Test Type, select the Wilcoxon test
Jamovi
Jamovi
Jamovi
Jamovi
Jamovi
Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
Put the two paired variables in the box below Measures
Regression > Correlation Matrix
Put your two variables in the white box at the right
Under Correlation Coefficients, select Pearson (selected by default)
Under Hypothesis, select your alternative hypothesis
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples  $\chi^2$ test of association
Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
Put your categorical variable in the box below Variable
Click on Expected Proportions and fill in the population proportions / probabilities according to $H_0$ in the boxes below Ratio. If $H_0$ states that they are all equal, you can leave the ratios equal to the default values (1)
TTests > Paired Samples TTest
Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
Under Tests, select Wilcoxon rank
Under Hypothesis, select your alternative hypothesis