Two sample t test  equal variances not assumed  overview
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Two sample $t$ test  equal variances not assumed
$z$ test for the difference between two proportions
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
Dependent variable
Dependent variable
Dependent variable
Dependent variable
One quantitative of interval or ratio level
One categorical with 2 independent groups
One of ordinal level
One categorical with 2 independent groups
Null hypothesis
Null hypothesis
Null hypothesis
Null hypothesis
H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.
H_{0}: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
H_{0}: the population median for group 1 is equal to the population median for group 2
Else:
Formulation 1:
H_{0}: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
Formulation 2:
H_{0}:
P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
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.
Model chisquared test for the complete regression model:
H_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
Wald test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
in the regression equation
$
\ln \big(\frac{\pi_{y = 1}}{1  \pi_{y = 1}} \big) = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K
$. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\pi_{y = 1}$ represents the true probability that the dependent variable $ y = 1$ (or equivalently, the proportion of $ y = 1$ in the population) given the scores on the independent variables.
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
H_{1} two sided: $\mu_1 \neq \mu_2$
H_{1} right sided: $\mu_1 > \mu_2$
H_{1} left sided: $\mu_1 < \mu_2$
H_{1} two sided: $\pi_1 \neq \pi_2$
H_{1} right sided: $\pi_1 > \pi_2$
H_{1} left sided: $\pi_1 < \pi_2$
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
H_{1} two sided: the population median for group 1 is not equal to the population median for group 2
H_{1} right sided: the population median for group 1 is larger than the population median for group 2
H_{1} left sided: the population median for group 1 is smaller than the population median for group 2
Else:
Formulation 1:
H_{1} two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2
H_{1} right sided: the population scores in group 1 are systematically higher than the population scores in group 2
H_{1} left sided: the population scores in group 1 are systematically lower than the population scores in group 2
Formulation 2:
H_{1} two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1)
H_{1} right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1)
H_{1} left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
Model chisquared test for the complete regression model:
H_{1}: not all population regression coefficients are 0
Wald test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
H_{1} right sided: $\beta_k > 0$
H_{1} left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
Assumptions
Assumptions
Assumptions
Assumptions
Within each population, the scores on the dependent variable are normally distributed
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 $z$ 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
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
In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1  \pi_{y=1}})$ is linear
The residuals are independent of one another
Often ignored additional assumption:
Variables are measured without error
Also pay attention to:
Multicollinearity
Outliers
Test statistic
Test statistic
Test statistic
Test statistic
$t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2,
$s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2,
$n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.
Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
$z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ 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, and $n_2$ 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 $\pi_2 > \pi_1.$
Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
The second type of test statistic is the MannWhitney $U$ statistic:
$U = W  \dfrac{n_1(n_1 + 1)}{2}$
where $n_1$ is the sample size of group 1.
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.
Model chisquared test for the complete regression model:
$X^2 = D_{null}  D_K = \mbox{null deviance}  \mbox{model deviance} $
$D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis.
Wald test for individual $\beta_k$:
The wald statistic can be defined in two ways:
Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$
Wald $ = \dfrac{b_k}{SE_{b_k}}$
SPSS uses the first definition.
Likelihood ratio chisquared test for individual $\beta_k$:
$X^2 = D_{K1}  D_K$
$D_{K1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.
Sampling distribution of $W$ and of $U$ if H_{0} were true
Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true
Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to
$k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1  1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2  1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$
or
$k$ = the smaller of $n_1$  1 and $n_2$  1
First definition of $k$ is used by computer programs, second definition is often used for hand calculations.
Approximately the standard normal distribution
Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here
$$
\begin{aligned}
\mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\
\sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}}
\end{aligned}
$$
Hence, for large samples, the standardized test statistic
$$
z_W = \dfrac{W  \mu_W}{\sigma_W}\\
$$
follows approximately the standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.
Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here
$$
\begin{aligned}
\mu_U &= \dfrac{n_1 n_2}{2}\\
\sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}}
\end{aligned}
$$
Hence, for large samples, the standardized test statistic
$$
z_U = \dfrac{U  \mu_U}{\sigma_U}\\
$$
follows approximately the standard normal distribution if the null hypothesis were true.
For small samples, the exact distribution of $W$ or $U$ should be used.
Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.
Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
chisquared distribution with $K$ (number of independent variables) degrees of freedom
Sampling distribution of the Wald statistic:
If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chisquared distribution with 1 degree of freedom
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution
Sampling distribution of $X^2$, as computed in the likelihood ratio chisquared test for individual $\beta_k$:
chisquared distribution with 1 degree of freedom
Significant?
Significant?
Significant?
Significant?
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 $\alpha$
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 $\alpha$
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 $\alpha$
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 $\alpha$
For large samples, 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 $\alpha$
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
For the Wald test:
If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chisquared tests. Wald can be interpret as $X^2$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.
Approximate $C\%$ confidence interval for $\mu_1  \mu_2$
Approximate $C\%$ confidence interval for $\pi_1  \pi_2$
n.a.
Waldtype approximate $C\%$ confidence interval for $\beta_k$
$(\bar{y}_1  \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).
$(p_1  p_2) \pm z^* \times \sqrt{\dfrac{p_1(1  p_1)}{n_1} + \dfrac{p_2(1  p_2)}{n_2}}$
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)

$b_k \pm z^* \times SE_{b_k}$
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).
n.a.
n.a.
n.a.
Goodness of fit measure $R^2_L$



$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$
There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit.
If there are no ties in the data, the two sided MannWhitneyWilcoxon test is equivalent to the KruskalWallis test with an independent variable with 2 levels ($I$ = 2).

Example context
Example context
Example context
Example context
Is the average mental health score different between men and women?
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.
Do men tend to score higher on social economic status than women?
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
SPSS
SPSS
SPSS
SPSS
Analyze > Compare Means > IndependentSamples T Test...
Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
Continue and click OK
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 dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
Continue and click OK
Analyze > Regression > Binary Logistic...
Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
Jamovi
Jamovi
Jamovi
Jamovi
TTests > Independent Samples TTest
Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
Under Tests, select Welch's
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
TTests > Independent Samples TTest
Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
Under Tests, select MannWhitney U
Under Hypothesis, select your alternative hypothesis
Regression > 2 Outcomes  Binomial
Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates
If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well
Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'