Two sample t test  equal variances not assumed  overview
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Two sample $t$ test  equal variances not assumed  One way ANOVA  Friedman test  Chisquared test for the relationship between two categorical variables  One sample $t$ test for the mean  KruskalWallis test  Marginal Homogeneity test / StuartMaxwell test  Chisquared test for the relationship between two categorical variables 


Independent/grouping variable  Independent/grouping variable  Independent/grouping variable  Independent /column variable  Independent variable  Independent/grouping variable  Independent variable  Independent /column variable  
One categorical with 2 independent groups  One categorical with $I$ independent groups ($I \geqslant 2$)  One within subject factor ($\geq 2$ related groups)  One categorical with $I$ independent groups ($I \geqslant 2$)  None  One categorical with $I$ independent groups ($I \geqslant 2$)  2 paired groups  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent variable  Dependent variable  Dependent /row variable  Dependent variable  Dependent variable  Dependent variable  Dependent /row variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  One quantitative of interval or ratio level  One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  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.  ANOVA $F$ test:
 H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
Usually the related groups are the different measurement points. 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.  H_{0}: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$  H_{0}: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  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: $\mu_1 < \mu_2$  ANOVA $F$ test:
 H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  H_{1}: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.  H_{1}: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  







 
Test statistic  Test statistic  Test statistic  Test statistic  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. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ 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 $\mu_2 > \mu_1$.  ANOVA $F$ test:
 $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.  
n.a.  Pooled standard deviation  n.a.  n.a.  n.a.  n.a.  n.a.  n.a.  
  $
\begin{aligned}
s_p &= \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2 + \ldots + (n_I  1) \times s^2_I}{N  I}}\\
&= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score}  \mbox{its group mean})^2}{N  I}}\\
&= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\
&= \sqrt{\mbox{mean square error}}
\end{aligned}
$
Here $s^2_i$ is the variance in group $i.$              
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $F$ and of $t$ if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $H$ if H_{0} were true  Sampling distribution of the test statistic if H_{0} were true  Sampling distribution of $X^2$ 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.  Sampling distribution of $F$:
 If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  $t$ distribution with $N  1$ degrees of freedom  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  Approximately the chisquared distribution with $J  1$ degrees of freedom  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  
Significant?  Significant?  Significant?  Significant?  Significant?  Significant?  Significant?  Significant?  
Two sided:
 $F$ test:
$t$ Test for contrast two sided:
$t$ Test multiple comparisons two sided:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:

 Two sided:
 For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 If we denote the test statistic as $X^2$:

 
Approximate $C\%$ confidence interval for $\mu_1  \mu_2$  $C\%$ confidence interval for $\Psi$, for $\mu_g  \mu_h$, and for $\mu_i$  n.a.  n.a.  $C\%$ confidence interval for $\mu$  n.a.  n.a.  n.a.  
$(\bar{y}_1  \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
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). The confidence interval for $\mu_1  \mu_2$ can also be used as significance test.  Confidence interval for $\Psi$ (contrast):
     $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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 $\mu$ can also be used as significance test.        
n.a.  Effect size  n.a.  n.a.  Effect size  n.a.  n.a.  n.a.  
 
     Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$        
Visual representation  n.a.  n.a.  n.a.  Visual representation  n.a.  n.a.  n.a.  
            
n.a.  ANOVA table  n.a.  n.a.  n.a.  n.a.  n.a.  n.a.  
 
Click the link for a step by step explanation of how to compute the sum of squares.              
n.a.  Equivalent to  n.a.  n.a.  n.a.  n.a.  n.a.  n.a.  
  OLS regression with one categorical independent variable transformed into $I  1$ code variables:
             
Example context  Example context  Example context  Example context  Example context  Example context  Example context  Example context  
Is the average mental health score different between men and women?  Is the average mental health score different between people from a low, moderate, and high economic class?  Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  Is the average mental health score of office workers different from $\mu_0 = 50$?  Do people from different religions tend to score differently on social economic status?  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  
SPSS  SPSS  SPSS  SPSS  SPSS  SPSS  SPSS  SPSS  
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