# Logistic regression - overview

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Logistic regression | Logistic regression | Two sample $t$ test - equal variances assumed |
You cannot compare more than 3 methods |
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Independent variables | 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 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 | Dependent variable | |

One categorical with 2 independent groups | One categorical with 2 independent groups | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

Model chi-squared test for the complete regression model:
- H
_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
| Model chi-squared test for the complete regression model:
- H
_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
| 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. | |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

Model chi-squared test for the complete regression model:
- H
_{1}: not all population regression coefficients are 0
- 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: $\beta_k < 0$
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$
| Model chi-squared test for the complete regression model:
- H
_{1}: not all population regression coefficients are 0
- 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: $\beta_k < 0$
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$
| H_{1} two sided: $\mu_1 \neq \mu_2$H _{1} right sided: $\mu_1 > \mu_2$H _{1} left sided: $\mu_1 < \mu_2$
| |

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
- Variables are measured without error
- Multicollinearity
- Outliers
| - 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
- Variables are measured without error
- Multicollinearity
- Outliers
| - Within each population, the scores on the dependent variable are normally distributed
- The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
- 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 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.
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}}$
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.
| 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.
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}}$
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.
| $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ 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 $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ 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 $\mu_2 > \mu_1$. | |

n.a. | n.a. | Pooled standard deviation | |

- | - | $s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$ | |

Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true | Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true | Sampling distribution of $t$ if H_{0} 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
- 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
- chi-squared distribution with 1 degree of freedom
| 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
- 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
- chi-squared distribution with 1 degree of freedom
| $t$ distribution with $n_1 + n_2 - 2$ degrees of freedom | |

Significant? | 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 $\alpha$
- 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 $z$.
| 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 $\alpha$
- 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 $t$ 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 $\alpha$
- 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 $\alpha$
- 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 $\alpha$
| |

Wald-type approximate $C\%$ confidence interval for $\beta_k$ | 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). | $b_k \pm z^* \times SE_{b_k}$ 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). | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test. | |

Goodness of fit measure $R^2_L$ | Goodness of fit measure $R^2_L$ | Effect size | |

$R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ 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. | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ 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. | Cohen's $d$:Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other. | |

n.a. | n.a. | Visual representation | |

- | - | ||

n.a. | n.a. | Equivalent to | |

- | - | One way ANOVA with an independent variable with 2 levels ($I$ = 2):
- two sided two sample $t$ test is equivalent to ANOVA $F$ test when $I$ = 2
- two sample $t$ test is equivalent to $t$ test for contrast when $I$ = 2
- two sample $t$ test is equivalent to $t$ test multiple comparisons when $I$ = 2
- two sided two sample $t$ test is equivalent to $F$ test regression model
- two sample $t$ test is equivalent to $t$ test for regression coefficient $\beta_1$
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Example context | Example context | Example context | |

Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | 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 deviation of mental health scores is equal amongst men and 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)
| Analyze > Regression > Binary Logistic...
- Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
| Analyze > Compare Means > Independent-Samples 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
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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 > 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'
| T-Tests > Independent Samples T-Test
- 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 Student's (selected by default)
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |