Parametric and Non-Parametric Tests of Independence

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Learning Module 9: Parametric and Non-Parametric Tests of Independence

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Pearson Correlation (or Bivariate Correlation)

\[ r_{XY} = \frac{s_{XY}}{s_X s_Y} \tag{1} \]

Where:

• \(r_{XY}\): Pearson correlation coefficient between variables \(X\) and \(Y\)
• \(s_{XY}\): sample covariance between \(X\) and \(Y\)
• \(s_X\): sample standard deviation of \(X\)
• \(s_Y\): sample standard deviation of \(Y\)

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## Pearson Correlation (or Bivariate Correlation)

$$
r_{XY} = \frac{s_{XY}}{s_X s_Y} \tag{1}
$$

Where:

* $r_{XY}$: Pearson correlation coefficient between variables $X$ and $Y$
* $s_{XY}$: sample covariance between $X$ and $Y$
* $s_X$: sample standard deviation of $X$
* $s_Y$: sample standard deviation of $Y$

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t-Test Statistic for Pearson Correlation

\[ t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} \tag{2} \]

Where:

• \(t\): test statistic for hypothesis testing of the correlation coefficient
• \(r\): sample Pearson correlation coefficient
• \(n - 2\) degrees of freedom
• \(n\): sample size

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## t-Test Statistic for Pearson Correlation

$$
t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} \tag{2}
$$

Where:

* $t$: test statistic for hypothesis testing of the correlation coefficient
* $r$: sample Pearson correlation coefficient
* $n - 2$ degrees of freedom
* $n$: sample size

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Spearman Rank Correlation

\[ r_s = 1 - \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 - 1)} \tag{3} \]

Where:

• \(r_s\): Spearman rank correlation coefficient
• \(d_i\): difference between the ranks of paired observations for item \(i\)
• \(n\): sample size

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## Spearman Rank Correlation

$$
r_s = 1 - \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 - 1)} \tag{3}
$$

Where:

* $r_s$: Spearman rank correlation coefficient
* $d_i$: difference between the ranks of paired observations for item $i$
* $n$: sample size

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Chi-Square Test Statistic for Independence

\[ \chi^2 = \sum_{i=1}^{m} \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \tag{4} \]

Where:

• \(\chi^2\): chi-square test statistic
• \(m\) = the number of cells in the table, which is the number of groups in the first class multiplied by the number of groups in the second class;
• \(O_{ij}\) = the number of observations in each cell of row \(i\) and column \(j\) (i.e., observed frequency); and
• \(E_{ij}\) = the expected number of observations in each cell of row \(i\) and column \(j\), assuming independence (i.e., expected frequency).

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## Chi-Square Test Statistic for Independence

$$
\chi^2 = \sum_{i=1}^{m} \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \tag{4}
$$

Where:

* $\chi^2$: chi-square test statistic
* $m$ = the number of cells in the table, which is the number of groups in the first class multiplied by the number of groups in the second class;
* $O_{ij}$ = the number of observations in each cell of row $i$ and column $j$ (i.e., observed frequency); and
* $E_{ij}$ = the expected number of observations in each cell of row $i$ and column $j$, assuming independence (i.e., expected frequency).

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Calculating Expected number of ETFs \((E_{ij})\)

\[ E_{ij} = \frac{(\text{Total row } i) \times (\text{Total column } j)}{\text{Overall total}} \tag{5} \]

Where:

• \(E_{ij}\): The expected number of ETFs
• \(\text{Total row } i\): sum of observed frequencies in row \(i\)
• \(\text{Total column } j\): sum of observed frequencies in column \(j\)
• \(\text{Overall total}\): total number of observations in the table

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## Calculating Expected number of ETFs $(E_{ij})$

$$
E_{ij} = \frac{(\text{Total row } i) \times (\text{Total column } j)}{\text{Overall total}} \tag{5}
$$

Where:

* $E_{ij}$: The expected number of ETFs
* $\text{Total row } i$: sum of observed frequencies in row $i$
* $\text{Total column } j$: sum of observed frequencies in column $j$
* $\text{Overall total}$: total number of observations in the table

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Standardized Residual (also referred to as a Pearson residual)

\[ \text{Standardized residual} = \frac{O_{ij} - E_{ij}}{\sqrt{E_{ij}}} \tag{6} \]

Where:

• \(O_{ij}\): the number of observations in each cell of row \(i\) and column \(j\) (i.e., observed frequency); and
• \(E_{ij}\) = the expected number of observations in each cell of row \(i\) and column \(j\), assuming independence (i.e., expected frequency).

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## Standardized Residual (also referred to as a Pearson residual)

$$
\text{Standardized residual} = \frac{O_{ij} - E_{ij}}{\sqrt{E_{ij}}} \tag{6}
$$

Where:

* $O_{ij}$:  the number of observations in each cell of row $i$ and column $j$ (i.e., observed frequency); and
* $E_{ij}$ = the expected number of observations in each cell of row $i$ and column $j$, assuming independence (i.e., expected frequency).

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