Reasoning in Psychology Using Statistics

Reasoning in PsychologyUsing Statistics Psychology 138 2015

Start working on your Final Projects soon (see link on syllabus page) • DueWed, April 29, in labs • Lab instructor assign a case in lab on Monday • Make sure to download: • 1) Your case datafile • 2) Directions • Write in sentences and paragraphs. Don’t just copy and paste SPSS; also interpret the output. • 3) Checklist (and attach to what you turn in) • Need to run SPSS • During lab after finish lab exercise or Milner lab or DEG 17 (PRC) • Check the PRC hours: http://psychology.illinoisstate.edu/prc/hours.shtml Final Projects

Changing focus • Looking for differences between groups: • ONE VARIABLE • Looking for relationships between TWO VARIABLES Decision tree

Describing the strength of the relationship Quantitative variables Two variables Relationship between variables • Changing focus • Looking for relationships between variables • (not looking for differences between groups) • Today’s topic: Pearson’s correlation Decision tree

study time • test performance 115 mins 15 mins • Relationships between variables may be described with correlation procedures Suppose that you notice that the more you study for an exam, the better your score typically is. • This suggests that there is a relationship between: Relationships between variables

Y 6 5 4 • Make a Scatterplot • Compute the Correlation Coefficient 3 2 1 X 1 2 3 5 4 6 • Determine whether the correlation coefficient is statistically significant - hypothesis testing New • Relationships between variables may be described with correlation procedures To examine this relationship you should: Relationships between variables

rcritical r = 1.0 perfect positive corr. r = -1.0 perfect negative corr. r = 0.0 no relationship -1.0 0.0 +1.0 The farther from zero, the stronger the relationship How strong a correlation to conclude it’s beyond what expected by chance? rcritical Review & New

Y 6 5 4 3 2 1 X 1 2 3 4 5 6 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

Step 1 Step 2 Step 3 • Step 1: compute Sum of the Products (SP) r = degree to which X and Y vary together • Step 2: SSX & SSY degree to which X and Y vary separately • Step 3: compute r • Pearson product-momentcorrelation • A numeric summary of the relationship Review: Computing Pearson’s r

2.4 2.0 4.8 4.0 5.76 -2.6 6.76 -2.0 4.0 5.2 1.4 1.96 2.0 4.0 2.8 -0.6 0.36 0.0 0.0 0.0 -0.6 0.36 -2.0 4.0 1.2 SP 3.6 4.0 0.0 15.20 0.0 16.0 14.0 SSX mean SSY Step 2 Step 1 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

Step 3 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 SP 15.20 16.0 14.0 SSX SSY Example Step 2 Step 1

Y 6 5 4 3 2 1 X 1 2 3 5 4 6 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? • Appears linear • Positive relationship • Fairly strong relationship • .898 is far from 0, near +1 A 6 6 B 1 2 C 5 6 • Fairly strong, but stronger than you’d expect by chance? D 3 4 E 3 2 Example

Hypothesis testing • Core logic of hypothesis testing • Considers the probability that the result of a study could have come about if no effect (in this case “no relationship”) • If this probability is low, then the scenario of no effect (relationship) is rejected Y 6 5 4 3 2 1 X 1 2 3 5 4 6 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? A 6 6 B 1 2 C 5 6 • Fairly strong, but stronger than you’d expect by chance? D 3 4 E 3 2 Example

Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data • Step 4: Compute your test statistics • Step 5: Make a decision about your null hypothesis A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations • Null hypothesis (H0) • Research hypothesis (HA) • There are no correlation between the variables (they are independent) • Generally, the variables correlated (they are not independent) Hypothesis testing with Pearson’s r

Step 1: Hypotheses One -tailed Hypothesize that variables are negatively correlated Note: symbol ρ(rho) is actually correct, but rarely used H0: ρ≥ 0 HA: ρ < 0 Hypothesis testing with Pearson’s r

Step 1: Hypotheses One -tailed Two -tailed Hypothesize that variables are negatively correlated Hypothesize that variables are correlated (either direction) H0: H0: ρ =0 ρ≥ 0 HA: ρ < 0 HA: ρ ≠ 0 Hypothesis testing with Pearson’s r

H0: ρ=0 HA: ρ ≠ 0 Suppose that you think that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05). • Step 1 2-tailed There is no correlation between the study time and exam performance A 6 6 B 1 2 C 5 6 There is a correlation between the study time and exam performance D 3 4 E 3 2 Example: New

Step 1: Hypotheses • Step 2: Criterion for decision • Alpha (α) level as guide for when to reject or fail to reject the null hypothesis. • Based on probability of making type I error Hypothesis testing with Pearson’s r

H0: ρ =0 HA: ρ ≠ 0 You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05). 2-tailed • Step 2 A 6 6 α = 0.05 B 1 2 C 5 6 D 3 4 E 3 2 Example: New

Step 1: Hypotheses • Step 2: Criterion for decision • Steps 3 & 4: Sample & Test statistics • Descriptive statistics (Pearson’s r) • Degrees of freedom (df): df = n – 2 • Used up one for each variable for calculating its mean • Note that n refers to number of pairs of scores, as in related-samples t-tests Hypothesis testing with Pearson’s r

Y 6 5 4 3 2 1 X 1 2 3 5 4 6 H0: ρ =0 HA: ρ ≠ 0 You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05). 2-tailed α = 0.05 A 6 6 • Steps 3 & 4 B 1 2 r = 0.898 C 5 6 df = n - 2 = 5 - 2 =3 D 3 4 E 3 2 Example: New

Critical values of r (rcrit) • Step 1: Hypotheses • Step 2: Criterion for decision • Steps 3 & 4: Sample & Test statistics • Step 5: Compare observed and critical test values • Use the Pearson’s r table (based on t-test or r to z transformation) Note: For very small df, need very large r for significance Hypothesis testing with Pearson’s r

Y 6 5 4 3 2 1 X 1 2 3 5 4 6 • rcrit = ±0.878 H0: ρ =0 HA: ρ ≠ 0 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? 2-tailed df = n - 2 = 3 α = 0.05 • Step 5 A 6 6 • From table B 1 2 C 5 6 D 3 4 E 3 2 Example: New

rcritical Fail to Reject H0 Reject H0 -1.0 0.0 +1.0 • Step 1: Hypotheses • Step 2: Criterion for decision • Steps 3 & 4: Sample & Test statistics • Step 5: Compare observed and critical test values • & Make a decision about H0 & Conclusions 1-tailed case when H0: r > 0 Hypothesis testing with Pearson’s r

Y 6 2-tailed 5 4 3 2 1 X 1 2 3 5 4 6 H0: r = HA: r ≠  -1.0 0.0 +1.0 The observed correlation is farther away from zero than the rcritical so we reject H0 Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score).Is there a statistically significant relationship between these variables (alpha = 0.05)? df = n - 2 = 3 alpha = 0.05 A 6 6 • rcrit = ±0.878 B 1 2 • Step 5 C 5 6 • Reject H0 • Conclude that the correlation is not equal to 0 D 3 4 E 3 2 “There is a significant positive correlation between study time and exam performance” Example: New

In labs: • Hypothesis testing with correlation (by hand and with SPSS) • Questions? Wrap up

Note that significance is expressed the same as previously r (38) = .794, p < .01 What is p for 1-tailed test? For df = 38, α = .05, 2-tailed, rcrit = .31 SPSS: HGT.SAVHeight by Weight, N = 40

Minimum N = 30, df = 28, rcrit = .30

Reasoning in Psychology Using Statistics