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$\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population.
In words:
there is no monotonic relationship between the two variables in the population
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:
The median in population 1 is equal to the median in population 2
Else:
Formulation 1:
The scores in population 1 are not systematically higher or lower than the scores in population 2
Formulation 2:
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.
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Model chisquared test for the complete regression model:
not all population regression coefficients are 0
Wald test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
right sided: $\beta_k > 0$
left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
Two sided: $\rho_s \neq 0$
Right sided: $\rho_s > 0$
Left sided: $\rho_s < 0$
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:
Two sided: the median in population 1 is not equal to the median in population 2
Right sided: the median in population 1 is larger than the median in population 2
Left sided: the median in population 1 is smaller than the median in population 2
Else:
Formulation 1:
Two sided: The scores in population 1 are systematically higher or lower than the scores in population 2
Right sided: The scores in population 1 are systematically higher than the scores in population 2
Left sided: The scores in population 1 are systematically lower than the scores in population 2
Formulation 2:
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)
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)
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)
Assumptions
Assumptions
Assumptions
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
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
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
Test statistic
Test statistic
Test statistic
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.
$t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $
where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
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.
Sampling distribution of $X^2$ and of the Wald statistic if H0 were true
Sampling distribution of $t$ if H0 were true
Sampling distribution of $W$ and of $U$ if H0 were true
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 a chisquared distribution with 1 degree of freedom
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately a 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
Approximately a $t$ distribution with $N  2$ degrees of freedom
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 a 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 a standard normal distribution if the null hypothesis were true.
For small samples, the exact distribution of $W$ or $U$ should be used.
Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.
Significant?
Significant?
Significant?
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 $z$.
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 $\alpha$
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 $\alpha$
Waldtype approximate $C\%$ confidence interval for $\beta_k$
n.a.
n.a.
$b_k \pm z^* \times SE_{b_k}$
where $z^*$ 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)


Goodness of fit measure $R^2_L$
n.a.
n.a.
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$
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.


n.a.
n.a.
Equivalent to


If no ties in the data: two sided MannWhitneyWilcoxon test is equivalent to KruskalWallis test with an independent variable with 2 levels ($I = 2$)
Example context
Example context
Example context
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
Is there a monotonic relationship between physical health and mental health?
Do men tend to score higher on social economic status than women?
SPSS
SPSS
SPSS
Analyze > Regression > Binary Logistic...
Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
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
Jamovi
Jamovi
Jamovi
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'
Regression > Correlation Matrix
Put your two variables in the white box at the right
Under Correlation Coefficients, select Spearman
Under Hypothesis, select your alternative hypothesis
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