Reasoning in Psychology Using Statistics 2019

Many students found the lab exam too long/difficult to complete Exam 3 extra credit opportunity (rather than a “curve”) for everybody (regardless of how you did on the exam) May earn up to 15 points of extra-credit This is a one time offer late submissions will not be accepted the due date is:?? Exam 3 Extra-credit

Topics remaining 2 ½ weeks of new content 3 more hypothesis tests Estimation Final Project Topics remaining

Start working on your Final Projects soon (see link on syllabus page) Due Wed, May 1 (uploaded to ReggieNet Assignment: Final Project) Lab instructor assign a case in lab Monday Make sure to download: Your case datafile Expectations Write in sentences and paragraphs. Don’t just copy and paste SPSS; also interpret the output. There is a “sample paper” provided. Checklist Need to run SPSS During lab after finish lab exercise or Milner lab or DEG 17 (PRC) PRC hours: https://psychology.illinoisstate.edu/prc/hours/ Final Projects

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

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

Relationships between variables 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: study time test performance 115 mins 15 mins Relationships between variables

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

Review & New -1.0 0.0 +1.0 Fail to Reject H0 Reject H0 rcritical perfect negative corr. r = 0.0 no relationship r = 1.0 perfect positive corr. Fail to Reject H0 Reject H0 rcritical -1.0 0.0 +1.0 The farther from zero, the stronger the relationship How strong a correlation to conclude it is beyond what expected by chance? Review & New

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)? Y X 1 2 3 4 5 6 X Y A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

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

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)? X Y A 6 6 2.4 -2.6 1.4 -0.6 0.0 5.76 6.76 1.96 0.36 15.20 2.0 -2.0 0.0 4.0 0.0 16.0 14.0 4.8 5.2 2.8 0.0 1.2 B 1 2 C 5 6 D 3 4 E 3 2 mean 3.6 4.0 SSY SSX Step 2 SP Step 1 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 3 X Y A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 15.20 SSX 16.0 SSY 14.0 SP Example Step 2 Step 1

Example Appears linear Positive relationship 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)? Y X 1 2 3 4 5 6 Appears linear Positive relationship Fairly strong relationship .898 is far from 0, near +1 X Y A 6 6 B 1 2 C 5 6 Fairly strong, but stronger than you would expect by chance? D 3 4 E 3 2 Example

Example Hypothesis testing 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)? 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 X 1 2 3 4 5 6 X Y A 6 6 B 1 2 C 5 6 Fairly strong, but stronger than you would expect by chance? D 3 4 E 3 2 Example

Example Step 1: State your hypotheses 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 X Y A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

Hypothesis testing with Pearson’s r 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) ρ = 0 Generally, the variables correlated (they are not independent) ρ ≠ 0 Note: symbol ρ (rho) is actually correct, but rarely used Hypothesis testing with Pearson’s r

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

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

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 X Y There is no correlation between the study time and exam performance A 6 6 B 1 2 ρ = 0 H0: C 5 6 There is a correlation between the study time and exam performance D 3 4 HA: ρ ≠ 0 E 3 2 Example: New

Hypothesis testing with Pearson’s r 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

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

Hypothesis testing with Pearson’s r 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

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

Hypothesis testing with Pearson’s r 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 Critical values of r (rcrit) Hypothesis testing with Pearson’s r

Example: New df = n - 2 = 3 Step 5 rcrit = ±0.878 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 ρ = 0 H0: HA: ρ ≠ 0 X Y df = n - 2 = 3 α = 0.05 Y X 1 2 3 4 5 6 A 6 6 Step 5 rcrit = ±0.878 From table B 1 2 C 5 6 D 3 4 E 3 2 Example: New

Hypothesis testing with Pearson’s r 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: ρ > 0 -1.0 0.0 +1.0 rcritical Fail to Reject H0 Reject H0 Hypothesis testing with Pearson’s r

r =  H0: HA: r ≠  2-tailed -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)? X Y df = n - 2 = 3 α = 0.05 A 6 6 rcrit = ±0.878 B 1 2 Y X 1 2 3 4 5 6 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

Generally, it is considered best to have at least 30 pairs of scores to conduct a Pearson’s r analysis Minimum N = 30, df = 28, rcrit = .30 Best Practice

Using Correlation in SPSS SPSS: HGT.SAV Height by Weight, N = 40 Note that significance is expressed the same as previously r (38) = .794, p < .001 What is p for 1-tailed test? For df = 38, α = .05, 2-tailed, rcrit = .31 Using Correlation in SPSS

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