Correlation Coefficients and SPSS Instructions

Correlation Coefficients

• Correlation coefficients provide two pieces of information:

  • Direction (positive or negative)
  • Strength (weak or strong)

• The further from zero (either positive or negative), the stronger the correlation.

• A correlation of zero is the weakest possible correlation.

• Example: $+0.3$ and $-0.3$ are equally strong correlations but have opposite directions.

Pearson vs. Spearman Correlation

• Pearson correlation is used when:

  • Data is on an interval scale.
  • The relationship is linear.

• Pearson correlation is preferred because it's a more sensitive measure.

• Spearman correlation is used when the relationship is nonlinear.

Interpretation of Correlation Coefficients

• Rules of thumb:

  • $0.3$: Weak correlation

  • $0.4$: Moderate correlation

  • $0.5$: Strong correlation

  • Above $0.5$ Very strong correlation

Examples of Correlation

• Father involvement vs. attachment:

  • Positive correlation (upward trend).
  • Pearson: $0.66$, Spearman: $0.68$ (very strong positive correlation).
  • Little $r$ is the symbol for Pearson correlation coefficient.
  • Symbols for Spearman: $r_s$, $rho$, or the Greek letter $\rho$ (rho).

• Tension vs. Easily overwhelmed:

  • Positive, but weaker correlation.
  • Pearson: $0.29$, Spearman: $0.28$ (positive weak correlation).

• Father involvement vs. troubled childhood:

  • Negative correlation.
  • Linear relationship (a straight line describes the data better than a curved line).
  • Monotonic relationship (line doesn't go up and down from left to right).
  • Pearson: $0.545$, Spearman: $0.48$ (moderate to strong negative correlation).

Computing Correlation Values

• Pearson $r$ formula (complex, never computed by hand).
  r = \frac{\sum{(xi - \bar{x})(yi - \bar{y})}}{\sqrt{\sum{(xi - \bar{x})^2 \sum{(yi - \bar{y})^2}}}
• Software like SPSS is used to compute correlations.

Cronbach's Alpha

• Measure of reliability (internal consistency) for items measuring the same construct.

  • Determines how consistent five items measuring the main variables all are.

• Steps to compute Cronbach's alpha in SPSS: (Analyze > Scale > Reliability Analysis).

• Move items measuring the main variable (e.g., anxiety) from left to right.

  • Do NOT include demographics or distractor items.

• Select "Alpha" in the drop-down menu.

• A Cronbach's alpha of $0.7$ or higher is considered good and reliable.

• If alpha is low, check for reverse coding errors.

• To improve alpha, analyze every combination of four items by removing one item at a time.

  • If removing an item increases alpha, the item is bad and should be removed.

  • If removing an item decreases alpha, the item is good and should be kept.

Creating a Measure of Anxiety

• In SPSS, go to (Transform > Compute Variable).
• Add the scores from the four (or five) good items.
• A new column with anxiety scores will be created in the dataset.

Computing Correlations in SPSS

• Go to (Analyze > Correlate > Bivariate).
• Move the variables of interest (e.g., depression and anxiety) to the right.
• Select Pearson and Spearman (or both) correlation.
• The output shows means, standard deviations, and correlation coefficients.
• The correlation coefficient is in the upper right-hand corner of the output.

Creating a Scatter Plot in SPSS

• Go to (Graphs > Scatter).

• Add a title (e.g., "Correlation between Depression and Anxiety").

• Decide which variable goes on the x-axis (cause) and y-axis (effect).

• If data points overlap, double-click on the scatter plot to open the editor.

• Go to (Options > Element) to show overlapping data points.

• Click on any of the dots, then go to Marker and increase the size (e.g., size 10) and change the color to black.

Interpreting Correlations (Review)

• Important information:

  • Direction (positive or negative).

  • Strength (weak, moderate, strong).

  • Correlation coefficients (r and rho) do not determine linearity or whether a relationship is monotonic.

Correlation vs. Causation

• Correlation does not imply causation, no matter how strong the correlation is.

Comparing Two Correlations

• To determine how much stronger one correlation is than another, compute $r^2$ (square the correlation coefficient).

• Example: If $r<em>1 = 0.2$ ($r^2 = 0.04$) and $r</em>2 = 0.4$ ($r^2 = 0.16$), then $r<em>2$ is four times as strong as $r</em>1$.