Quiz 5: Bivariate Correlation and Regression Analysis

What is the definition of correlation in the context of interval-level variables?

A measure of both the association and the direction of the relationship between two variables.

Which graphic device is primarily used to gauge the association between two interval variables?

A scatter plot.

As points in a scatter plot more closely form an imaginary diagonal line, what happens to the strength of the correlation?

The strength of the correlation increases.

How is a positive correlation defined in terms of variable scores?

High scores on the $X$ variable are associated with high scores on the $Y$ variable.

How is a negative correlation defined in terms of variable scores?

High scores on the $X$ variable are associated with low scores on the $Y$ variable.

What type of relationship is represented by a positive or negative correlation that forms a straight line?

A linear relationship.

What term describes a relationship between $X$ and $Y$ that changes direction, such as starting as positive and becoming negative?

A curvilinear relationship.

What is the specific name of the correlation coefficient used for linear relationships between two interval-level variables?

Pearson's correlation coefficient ($r$).

What is the numerical range of the Pearson's correlation coefficient?

$-1.0$ to $+1.0$.

In a correlation coefficient, what value represents total independence or no association between variables?

0

Pearson's $r$ focuses on the product of the deviations of $X$ and $Y$ from what specific values?

Their respective means.

What is the definitional formula for Pearson's $r$ using Sum of Products ($SP$) and Sum of Squares ($SS$)?

$r = \frac{SP}{\sqrt{SS_X SS_Y}}$

When testing the significance of Pearson's $r$, what does the null hypothesis state about the population correlation ($\rho$)?

The population correlation is zero ($\rho = 0$).

What is the formula for the degrees of freedom used to test the significance of Pearson's $r$?

$df = N - 2$

What is the formula for the $t$ ratio used to test the significance of Pearson's $r$?

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

What is the definition of a partial correlation coefficient?

The correlation between two variables after removing the common effects of a third variable.

Which statistic is useful for determining if the relationship between two variables is spurious?

The partial correlation coefficient.

What is one major advantage of regression analysis over simple correlation regarding causal relationships?

It can be used to ascertain causal relationships.

In the simple bivariate regression equation $Y = a + bX + e$, what does the symbol '$a$' represent?

The $Y$ intercept (the expected value of $Y$ when $X = 0$).

In the regression equation $Y = a + bX + e$, what does the symbol '$b$' represent?

The slope or regression coefficient.

In regression analysis, what does the slope ($b$) represent regarding the change in $Y$?

The amount $Y$ is expected to change for each increase of one unit in $X$.

What does the existence of the error term ($e$) in a regression model imply about the relationship between $X$ and $Y$?

It implies that $X$ cannot completely explain $Y$.

What is the goal of the Ordinary Least Square (OLS) method in regression analysis?

To estimate the regression coefficients.

What are the two primary tasks to be accomplished in a regression analysis?

Estimate population parameters and test the statistical significance of the model/coefficients.

What are the requirements for simple classical regression regarding measurement level?

Both variables should be measured at the interval level.

In regression, if $b < 0$, how are the independent and dependent variables related?

They are negatively related.

What is the mathematical definition of the error or disturbance term ($e$)?

$e = Y - \hat{Y}$ (the difference between the actual $Y$ and the predicted $\hat{Y}$).

Which sum of squares represents the variations in $Y$ that can be explained by $X$?

Regression sum of squares ($SS{reg}$).

Which sum of squares represents the remaining variations in $Y$ that are unexplained by $X$?

Error sum of squares ($SS{error}$).

What does the Proportionate Reduction in Error (PRE) statistic indicate in regression?

How much of the variance in the dependent variable can be explained by the independent variable.

What is the formula for the PRE statistic using Sum of Squares?

$PRE = \frac{SS{total} - SS{error}}{SS{total}}$

What is the specific name for the $r^2$ value in regression analysis?

The coefficient of determination.

For bivariate regression, the coefficient of determination ($r^2$) is equal to the square of which other statistic?

Pearson's correlation coefficient ($r$).

What does the coefficient of non-determination ($1 - r^2$) measure?

The proportion of variance in $Y$ that is not explained by $X$.

How does multiple regression differ from simple regression regarding independent variables?

Multiple regression considers two or more independent variables simultaneously.

What is the primary purpose of using multiple regression over simple regression?

It allows control for the effects of more than one independent variable on the dependent variable.

In multiple regression, what does each regression coefficient represent?

The net effect of a particular variable after the effects of all other independent variables have been controlled.

What are 'dummy variables' in the context of regression analysis?

Nominal variables converted into dichotomous special variables coded as 0 and 1.

What is an interaction effect in a regression model?

The joint effect of two or more independent variables on the dependent variable in addition to their main effects.

Which statistical method is preferred for predicting a dichotomous dependent variable (constrained to 0 or 1)?

Logistic regression (or probit regression).

Logistic regression models the natural logarithm of the odds, which is symbolized by $L$ and known as the __.

Logit (or log-odds).

Under what condition should a researcher use nonparametric measures of association?

When using nominal or ordinal data, or if the assumption of normality cannot be met.

Which nonparametric correlation coefficient is used specifically for two variables measured at the ordinal level?

Spearman's rank-order correlation coefficient ($r_s$).

When ordinal variables are cross-tabulated and contain many tied ranks, which measure of association is particularly useful?

Goodman's and Kruskal's gamma ($G$).

What two components are used to calculate Goodman's and Kruskal's gamma ($G$)?

The number of agreements ($N_a$) and the number of inversions ($N_i$).

Which correlation coefficient is designed for nominal data arranged in a $2 \times 2$ table?

The phi coefficient ($\phi$).

The phi coefficient is based on an extension of which statistical test?

The chi-square test.

Which measure of association is used for nominal variables when either variable has more than two categories?

The contingency coefficient ($C$).

What is the main disadvantage of the contingency coefficient ($C$)?

Its maximum size is influenced by the number of rows and columns in the chi-square table.

Which statistic is used to avoid the limitations of the contingency coefficient regarding table size?

Cramér's $V$.

What distinguishes 'parametric' tests (like $t$ tests) from 'nonparametric' tests (like chi-square)?

Parametric tests assume population parameters to be estimated and require normality and interval-level data.

In a Chi-square test, what does $f_o$ represent?

The observed frequency (the actual counted distribution).

In a Chi-square test, what does $f_e$ represent?

The expected frequency (the distribution anticipated based on the null hypothesis).

What is the formula for the degrees of freedom in a one-way Chi-square test?

$df = k - 1$ (where $k$ is the number of categories).

What is the primary use of a two-way Chi-square test?

To determine whether two categorical variables are associated or independent.

How are expected frequencies ($f_e$) calculated for a specific cell in a contingency table?

$f_e = \frac{(\text{row marginal total}) \times (\text{column marginal total})}{N}$

What is the formula for degrees of freedom in a two-way Chi-square test?

$df = (r - 1)(c - 1)$

What is the general rule of thumb for expected cell frequencies to ensure a Chi-square test is valid?

Expected frequencies should not be less than five.

When should Yate's correction be applied to a Chi-square test?

In $2 \times 2$ tables when cell frequencies are small (typically between 5 and 10).

Which nonparametric test is designed specifically for comparing two or more groups using ordinal data?

The Median test.

In a Median test, the observed frequencies are based on the number of cases falling into which two categories?

Above the median and below the median.

What is the formula for Pearson's $r$ in terms of variance and covariance?

$r = \frac{\sigma{XY}^2}{\sigma_X \sigma_Y}$

Which sum of squares calculation represents the total variation of $Y$?

$SS{total}$

If the calculated Pearson's $r$ exceeds the critical value in Table F, what action should the researcher take regarding the null hypothesis?

Reject the null hypothesis.

Concept: $SP$ (Sum of Products)

The sum of the products of the $X$ and $Y$ deviations from their means used in the calculation of Pearson's $r$.

In the regression equation $\hat{Y} = a + bX$, what does $\hat{Y}$ represent?

The predicted value of $Y$.

What is the relationship between $SS{reg}$ and $r^2$?

$SS{reg} = r^2 \times SS{total}$

What is the relationship between $SS{error}$ and $r^2$?

$SS{error} = (1 - r^2) \times SS{total}$

In Spearman's rank-order correlation formula, what does $D$ represent?

The difference in rank between the $X$ and $Y$ variables for a specific case.

What is the formula for converting Spearman's $r_s$ to a $z$ score for samples where $N > 30$?

$z = r_s \sqrt{N - 1}$

In Goodman's and Kruskal's Gamma, what is an 'agreement' ($N_a$)?

A pair of cases ranked in the same order on both variables.

In the calculation of Cramér's $V$, what does the value '$k$' represent?

The number of rows or columns, whichever is smaller.

What is the primary assumption about the independent variable in a classical regression model?

It is measured at the interval level (though this assumption can be relaxed).

The formula $b = \frac{SP}{SS_X}$ is used to calculate what?

The slope (regression coefficient) in an OLS simple regression.

What is the relationship between the slope $b$ and the Pearson's correlation $r$?

$r = \frac{s_x}{s_y} b$

If $b = 0$ in a regression analysis, what is the conclusion about the relationship between $X$ and $Y$?

$X$ and $Y$ are unrelated.

Term: Odds Ratio

The odds of an event occurring for one group divided by the odds for another group, used in logistic regression.

Which sum of squares is equivalent to the 'between-group' sum of squares in ANOVA within the context of regression?

The regression sum of squares ($SS{reg}$).

How are tied ranks handled when calculating Spearman's $r_s$?

Sum the tied ranks and divide by the number of tied cases.