Statistics

Z-Scores and Standard Normal Distribution

• Z-scores are a transformation of raw scores into units of standard deviation
• A z-score of 1 means the raw score is 1 standard deviation above the mean, and a z-score of -1 means the raw score is 1 standard deviation below the mean
• The distribution of z-scores should have the same shape as the distribution of raw scores, normally a normal distribution
• The formula for transforming raw scores into z-scores is: (Raw score - mean of distribution) / SD of distribution
• The standard normal distribution describes the normal distribution of z-scores

Using Z-Scores and Standard Normal Distribution

• Z-scores can be used to know the percentage of scores below each value
• Z-scores can be used to compare individual scores on one variable with scores on another variable
• Z-scores can be used to know which score on a variable sits at a specific percentile
• The formula for transforming a z-score back into a raw score is: X = (z) (Sx) + X

Correlation

• A positive association is a type of correlation where if one variable increases or decreases, the other does the same
• A negative association is a type of correlation where if one variable increases or decreases, the other does the opposite
• An undefined association is a type of correlation where scores are organized horizontally on a scatter plot
• A perfect association is a type of correlation that can be either positive or negative, where the change in one variable is exactly proportional to the change in the other
• Correlation can be assessed pictorially using a scatter plot or numerically using a correlation coefficient

Scatter Plots

• A scatter plot can be used to set out where each participant lies on the scale of each variable, compared to other participants
• Scatter plots assess the direction of the relationship between variables, the strength of the relationship, and whether it is a linear relationship
• If the line of participant scores goes diagonally upwards from the bottom left to the top right, then it is a positive correlation. If the line goes diagonally downwards from the top left to the bottom right, then it is a negative correlation
• The more points are clustered close to the line of best fit, the stronger the relationship is

Correlation Coefficient

• Pearson's r is used on interval data and indicates the correlative relationship on a scale of -1 (perfect negative relationship) to +1 (perfect positive relationship), with 0 indicating no relationship
• Pearson's r is calculated by converting raw scores to z-scores, multiplying the z-scores for each participant, adding the products together, and dividing by the number of participants minus 1
• In SPSS output, the value of Pearson's r is displayed in the Correlations table, and the statistical significance of this value is displayed in the Sig. (2-tailed) row for each variable

Normality

• Normality consists of two components: skewness and kurtosis
• Skewness is the symmetry of a distribution chart, with a bell-curve having no skew
• Kurtosis is how sharply peaked a sample's distribution is
• Skewness and kurtosis can affect data analysis, but a larger sample size can reduce this risk

Factors Affecting Correlation Coefficient

• Inverted-U shaped relationships, restricted ranges, outliers, and the shape of X and Y distributions can all cause problems in the calculation of Pearson's r
• An inverted-U shaped relationship can underestimate the size of the possible correlation
• A restricted range can make the correlation appear reduced or inflated
• Outliers can strongly influence the Pearson's r calculations
• The shape of X and Y distributions can also affect the correlation coefficient

Regression Analysis

• Regression analysis involves finding the best-fitting straight line through a scatter plot
• The least squares criterion states that the line of best fit should be the line with the lowest possible sum of squared residuals
• The regression coefficient (b) is the number of units that the line moves up the Y axis for each unit it moves along the X axis
• The regression constant (a) is the point value of the place where the regression line meets the Y axis
• The least squares criterion is satisfied when b is equal to the sample of covariance of x and y divided by the sample variance of x

Multiple Regression

• Multiple regression involves more than one predictor variable
• The equation for multiple regression is: y = a + (b1)(x1) + (b2)(x2) + ...
• Hierarchical multiple regression is used when the researcher believes that the effects of the predictor on the outcome are not fully explained without the inclusion of the external predictor
• In SPSS output, the goodness-of-fit is found in the ANOVA table, the overall effects of the predictor variables are found in the model summary table, and the effects of each predictor variable (separately) are found in the coefficients table### Outliers and Residuals
• More than 5% of samples with a residual below -2.0 or above +2.0 indicate cause for concern.
• Cook's distance determines the predicted scores for other cases if a specific case is not included in the analysis.
• Distance scores above 1 indicate cause for concern.
• Cook's distance can be found in the Maximum column of the Residual Statistics table in SPSS output.

Addressing Outliers

• First, ensure outliers are not caused by data entry errors.
• Options for addressing outliers: transforming the data, or deleting the case responsible for the outlier (if it produces a large distortion).

Assumptions of Residuals in Regression Analysis

• Normality: residual values should be normally distributed.
• Linearity: residuals should have a straight line relationship with the predicted outcomes.
• Homoscedasticity: variance of the residuals should be approximately equal for all predicted scores.
• Independence of error: errors of prediction should be uncorrelated.

Checking Assumptions

• Normality, linearity, and homoscedasticity can be checked by inspecting the scatterplot of standardized residuals.
• A histogram can be used to check the normality of residuals.
• Independence of error can be investigated using the Durbin-Watson test.

Violating Assumptions

• If an assumption is severely violated, the regression should be re-run using robust methods (e.g., bootstrapping).

Multicollinearity

• Multicollinearity occurs when predictor variables in the model are highly correlated (>0.80).
• Multicollinearity becomes a problem when the research aims to find the separate effects of different predictors on the outcome.
• It can be checked by inspecting the correlation matrix.

Dealing with Multicollinearity

• One or more variables can be deleted from the model if the correlation between two predictors is very high (>0.90).
• Collinear variables can be combined into one composite variable if the correlation is around 0.80.

Mediation Analysis

• Mediation analysis tests whether the effect of the predictor on the outcome exists because of another variable.
• It reveals whether the third variable is relevant by reducing the effects of the predictor on the outcome.

Mediation Analysis Components

• Indirect effects: the effects of the predictor on the outcome through the mediator variable.
• Path a: the effect of the predictor on the mediator.
• Path b: the effect of the mediator on the outcome.
• Path c: the effect of the predictor on the outcome.

Calculating Indirect Effects

• The standardised coefficient for the indirect effect is calculated by multiplying the regression coefficients for path a and path b.

Moderation Analysis

• Moderation analysis establishes whether the relationship between the predictor and the outcome changes as a function of the level of a third variable (the moderator).

Moderation Analysis Components

• The outcome is predicted by the predictor variable, the proposed moderator, and the interaction of the two.
• The interaction effect indicates whether moderation has occurred.

Displaying Moderation Analysis Results

• A simple slope graph is used to illustrate the effect of the predictor, the effect of the moderator, and the interaction/moderation effect.

PROCESS and Moderation Analysis

• PROCESS outputs do not provide standardized regression coefficients when an interaction is present.
• PROCESS requires complete data and a continuous outcome variable for moderation analysis.