Logistic regression overview

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Logistic regression	Two sample $z$ test	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent variables	Independent/grouping variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables	One categorical with 2 independent groups
Dependent variable	Dependent variable
One categorical with 2 independent groups	One quantitative of interval or ratio level
Null hypothesis	Null hypothesis
Model chi-squared test for the complete regression model: H₀: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$ Wald test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ 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.	H₀: $\mu_1 = \mu_2$ Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.
Alternative hypothesis	Alternative hypothesis
Model chi-squared test for the complete regression model: H₁: not all population regression coefficients are 0 Wald test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested: H₁ right sided: $\beta_k > 0$ H₁ left sided: $\beta_k < 0$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$	H₁ two sided: $\mu_1 \neq \mu_2$ H₁ right sided: $\mu_1 > \mu_2$ H₁ left sided: $\mu_1 < \mu_2$
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	Within each population, the scores on the dependent variable are normally distributed Population standard deviations $\sigma_1$ and $\sigma_2$ are known 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
Model chi-squared 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 chi-squared test for individual $\beta_k$: $X^2 = D_{K-1} - D_K$ $D_{K-1}$ 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.	$z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$ Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 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. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ is removed from 0. 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 $\mu_2 > \mu_1$.
Sampling distribution of $X^2$ and of the Wald statistic if H₀ were true	Sampling distribution of $z$ if H₀ were true
Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model: chi-squared 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 chi-squared 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 chi-squared test for individual $\beta_k$: chi-squared distribution with 1 degree of freedom	Standard normal distribution
Significant?	Significant?
For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or 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 chi-squared 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 $z$ 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 $\alpha$
Wald-type approximate $C\%$ confidence interval for $\beta_k$	$C\%$ confidence interval for $\mu_1 - \mu_2$
$b_k \pm z^* \times SE_{b_k}$ 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).	$(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$ 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). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.
Goodness of fit measure $R^2_L$	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.	Visual representation
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Example context	Example context
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?	Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women.
SPSS	n.a.
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	n.a.
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'	-
Practice questions	Practice questions

Logistic regression - overview