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 | McNemar's test | Friedman test |
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Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | Independent variable | Independent/grouping variable | |
One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | One within subject factor ($\geq 2$ related groups) | 2 paired groups | One within subject factor ($\geq 2$ related groups) | |
Dependent variable | Dependent variable | Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One quantitative of interval or ratio level | One of ordinal level | One categorical with 2 independent groups | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
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. | ANOVA $F$ test:
| H0: 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. | Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
| H0: 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. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | ANOVA $F$ test:
| H1: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups | The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
| H1: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups | |
Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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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. | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
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. | |
n.a. | Pooled standard deviation | 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 H0 were true | Sampling distribution of $F$ and of $t$ if H0 were true | Sampling distribution of $Q$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $Q$ if H0 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 chi-squared distribution with $k - 1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used. | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | If the number of blocks $N$ is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used. | |
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 $X^2 = Q$:
| For test statistic $X^2$:
| If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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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. | 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):
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n.a. | Effect size | n.a. | n.a. | n.a. | |
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Visual representation | n.a. | n.a. | n.a. | n.a. | |
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n.a. | ANOVA table | n.a. | n.a. | n.a. | |
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Click the link for a step by step explanation of how to compute the sum of squares. | - | - | - | |
n.a. | Equivalent to | n.a. | Equivalent to | n.a. | |
- | OLS regression with one categorical independent variable transformed into $I - 1$ code variables:
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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 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | Is there a difference in depression level between measurement point 1 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)? | |
SPSS | SPSS | SPSS | SPSS | SPSS | |
Analyze > Compare Means > Independent-Samples T Test...
| Analyze > Compare Means > One-Way ANOVA...
Analyze > General Linear Model > Univariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
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Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | |
T-Tests > Independent Samples T-Test
| ANOVA > ANOVA
| ANOVA > Repeated Measures ANOVA - Friedman
| Frequencies > Paired Samples - McNemar test
| ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |