CH 12 PSY 110

I. Overview
(A) Analysis of Variance (ANOVA): A procedure used to evaluate the differences between two or more
treatments (groups) – a t-test compares only two groups.
(B) Example: IV = Drug Dosage (0mg, 10mg, 20 mg)
II. Statistical Hypotheses for ANOVA
(A) H0: μ1 = μ2 = μ3 (assuming there are 3 groups – more μ values if there are more groups)
(B) H1: not H0
III. The Test Statistic for ANOVA
(A) Conceptual idea: F = variance (differences) between sample means
variance (differences) due to chance (error)
(B) Variance
(1) Why variance?  when there are more than 2 sample means it is impossible to compute
mean differences (therefore need to use “variance”)
(2) What is variance?  variance is simply a method for measuring how big the differences are
for a set of numbers (variance = differences)
IV. The Logic of ANOVA
(A) Example data:
(1) Not all the scores are the same – there is variability (variance)
(2) There are 3 types of variability – (a) total, (b) between-treatments, and (c) within-treatments
(B) Total Variability
(1) The deviation of each score from the Grand Mean (viz. the average of all the scores)
(2) This variability can be partitioned or divided into 2 parts:
(a) the amount that score deviates from its group mean
(i.e., within-treatment variation)
(b) the amount that score’s group mean differs from the grand mean
(i.e., between-treatment variation)

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(3) Example
0 mg 10 mg 20 mg
16 4 2
18 6 10
10 8 9
12 10 13
19 2 11
M1 = 15 M2 = 6 M3 = 9 MG (grand mean) = 10
Deviation from Grand Mean = Deviation from group mean + Deviation of mean from grand mean
(Total) (within-treatment) (between-treatment)
??? = ??? + ???
(C) Between-Treatments Variability
(1) Measures the differences between sample means
(2) What causes those differences between means?
(a) Treatment Effects: Systematic differences between groups due to the effect of the IV
(b) Chance or Experimental Error
(3) Sources of chance or experimental error
(a) Individual Differences (e.g., sleep, past drug use, IQ, etc.)
(b) Measurement Error (e.g., not calibrating a machine, misreading a dial, etc.)
(D) Within-Treatments Variability
(1) Logic  If all Ss were the same (no experimental error), all the subjects who received the
same treatment (e.g., 10 mg of a drug) would have the same score on the DV.
(2) Conclusion  The variability of subjects in the same condition gives us an estimate of
chance or experimental error. As such, within-treatments variability is solely
caused by chance or experimental error.
V. The F-ratio (the test statistic for ANOVA)
(A) Goal of ANOVA: To test for treatment effects (did the IV impact the DV). The F-ratio allows you
to accomplish this goal.
(B) F-ratio
(1) You form a ratio in which F = variance between treatments
variance within treatments
[“conceptual formula”] F = (treatment effects) + (experimental error) (between)
(experimental error) (within)
(C) If Ho is true, then there are no treatment effects and the “conceptual formula” reduces to:
(experimental error) (This equals 1)
(experimental error)
(D) If Ho is false, then there are treatment effects and the numerator or the “conceptual formula” gets
bigger and bigger (making the overall fraction large). If the fraction (i.e.,
value of this test statistic) gets large enough, we will feel that we have
adequate evidence to Reject Ho and conclude that we have treatment effects.

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VI. ANOVA Notation and Formulas
(A) Notation
k = the number of treatment conditions (groups)
n = the number of scores in each treatment
N = the total number of scores in the entire study
T = the total for each treatment condition (∑X) [needed for formulas so memorize]
G = the sum of all the scores in the study G = ∑T [needed for formulas so memorize]
(“Grand Total”)
(B) Total Sums of Squares (SSTotal) [Total Variability]
SSTotal = ∑X2 – (G2 / N)
** Using Table above  SSTotal = ???
(C) Within-Treatments Sums of Squares (SSWithin-treatments) [Within-Treatments Variability]
SSWithin-treatments = ∑SSinside each treatment (add up all SS values for each group)
** Using Table above  SSWithin-treatments = ???
(D) Between-Treatments Sums of Squares (SSBetween-treatment) [Between-Treatments Variability]
SSBetween-treatments = ∑(T2 / n) – (G2 / N)
** Using Table above  SSBetween-treatments = ???
CHECK: SSTotal = SSWithin-treatments + SSBetween-treatments
??? = ??? + ???
(E) Degrees of freedom for ANOVA
(1) dfTotal = N – 1 Using Table  dfTotal = ???
(2) dfWithin-treatments = N – k Using Table  dfWithin-treatments = ???
(3) dfBetween-treatments = k – 1 ** Using Table  dfBetween-treatments = ???
CHECK: Total = Within + Between
??? = ??? + ???
VII. Calculation of Mean Squares (MS) and the F-ratio
(A) Why do we Need Mean Squares?
-- Need to convert Sums of Squares (SS) into Variance estimates [i.e., Mean Square (MS)
values] to use them in Analysis of Variance

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(B) What is a Mean Square?
-- MS or Variance = SS/df
(C) MS Calculations:
(1) MSBetween-treatments = SSBetween-treatments / dfBetween-treatments
** Using Table  MSBetween-treatments = ???
(2) MSWithin-treatments = SSWithin-treatments / dfWithin-treatments
** Using Table  MSWithin-treatments = ???
(D) F-ratio Calculation
F = MSBetween-treatments / MSWithin-treatments
** Using Table  F = ???
VIII. The Distribution of F-ratios
(A) Characteristics of the F-distribution
(1) F-values will always be positive (there is never a negative variance)
(2) When H0 is true, F is close to 1.00. As such, the distribution piles up around 1.00.
(B) F-distribution Table (Table B.4 on page. 590)
(1) Purpose: Used to obtain critical value for boundary of your Critical Region
(2) Example: Using Table dfBetween-treatments = ??? (numerator of F)
dfWithin-treatments = ??? (denominator of F)
Fcritical (α = .05) = ???
** Note: Draw Figure
IX. Hypothesis Testing Example
* Research Question: Are these drugs equally effective as pain relievers?
* DV: # of seconds that a painful stimulus is endured
* Data:
(A) Step #1 – State the hypotheses
(1) H0: ???
(2) H1: ???
(B) Step #2 – Set the criteria for a decision
(1) α = .05
(2) Calculate df values dfTotal = N – 1  ???
dfBetween-treatments = k – 1  ???
dfWithin-treatments = N – k  ???
(3) Determine and draw boundary of the critical region
** F = ??? (from Table B.4)
** Draw graph with the critical regions

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(C) Step #3 – Collect data and compute sample statistics
(1) Part 1 = calculate SS values
(a) SSTotal = ∑X2 – (G2 / N)  ???
(b) SSWithin-treatments = ∑SSinside each treatment  ???
(c) SSBetween-treatments = ∑(T2 / n) – (G2 / N)  ???
(2) Part 2 = calculate MS values
(a) MSBetween-treatments = SSBetween-treatments / dfBetween-treatments  ???
(b) MSWithin-treatments = SSWithin-treatments / dfWithin-treatments  ???
(3) Part 3 = calculate F statistic
F = MSBetween-treatments / MSWithin-treatments  ???
(4) Part 4 = complete the ANOVA Summary Table
Source SS df MS F
Between treatments
Within treatments
Total
(D) Step #4 – Make a decision
(1) Draw obtained score (“test statistic”) on the graph (e.g., 8.33)
(2) Decision Rule:
(a) If obtained value is in the critical region, you “Reject H0”
(b) If obtained value is not in the critical region, you “Fail to Reject H0”
(3) Decision = ???
Note: Still need to do post-hocs to determine “what differs from what”
X. Assumptions of the 1-way Between subjects ANOVA
(A) Independent scores
(B) Normality (the populations from which the samples are selected must be normal)
(C) Homogeneity of Variance (the populations from which the samples are selected must
have equal variances)
XI. Post-hoc Tests
(A) Issue  After a significant F-test you still need to determine “what differs from what”
(B) Solution  Use Post-hoc tests
(1) Definition: Additional hypothesis tests that are done after an ANOVA to determine which
mean differences are significant and which are not.
(2) Strategy  Make pairwise comparisons (tests between 2 groups)
(C) When you use Post-hoc tests
(1) When you reject H0 and...
(2) There are three or more groups (don’t need it with two groups)

I. Introduction to Correlation
(A) Definition: Correlation is a statistical technique that is used to measure and describe a relationship
between two variables. These variables are simply observed – there is no attempt to
control or manipulate them (it is not an experiment).
(B) Example: # of days of sun and depression rates
(C) Display in a table or a scatterplot
II. Characteristics of a Relationship between Two Variables
(A) Direction of the relationship
(1) Positive correlation  the variables move in the same direction
(e.g., heat and aggression) (Note: draw this on the board)
(2) Negative correlation  the variables move in the opposite direction
(e.g., # of days of sun and depression rates) (Note: draw this on the board)
** see Figure 14.2 (on page 6 of lecture)
Note: The sign of the correlation value indicates the direction (+ = positive; - = negative)
(B) Form of the relationship
(1) Linear relationship (e.g., temperature and beer sales)
(2) Non-linear relationship (e.g., arousal and performance – Yerkes-Dodson Curve)
(C) Degree (strength) of the relationship
(1) How well the data fit the form or line
(2) Measured by the numerical value of the correlation
* Range of the correlation is -1.00 to +1.00
* Correlations of -1.00 and +1.00 are a perfect correlation
* Correlation of zero means no fit at all
III. The Pearson Correlation
(A) Definition: The Pearson correlation measures the degree and direction of the linear relationship
between two continuous variables.
(B) Range: -1.00 to +1.00
(C) Symbol: r

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IV. Creating the Formula for the Pearson Correlation
(A) The sum of products of deviations [Step #1 of Pearson formula]
(1) Symbol = SP
(2) Formula: SP = ∑XY – [(∑X ∑Y) / n]
(B) Complete Pearson Correlation Formula
r = SP / √SSXSSY (note: the whole denominator is in sqrt)
[Step #2 of Pearson formula]
(C) Example calculation
(1) Data X Y XY
0 1
10 3
4 1
8 2
8 3
∑X = ?? ∑Y = ?? ∑XY = ??
(2) Draw Scatterplot Note: ask class about (a) direction & (b) strength of relationship
(3) Calculate SP SP = ∑XY – [(∑X ∑Y) / n]  ???
(4) Calculate r r = SP / √SSXSSY  14 / √(64)(4)  ???
V. Hypothesis Testing with the Pearson Correlation
(A) The hypotheses
(1) H0: ρ = 0 (there is no population correlation) (assuming a 2-tailed test)
(2) H1: ρ ≠ 0 (there is a real population correlation)
(B) Degrees of Freedom
df = n – 2 (note: ask students to memorize this)
(C) Table of Critical Values
(1) Table B.6 (Note: Go over this table with the class – p. 601)
* Example: if sample is n=30 and α=.05 and two-tailed  rcrit = ???
* Draw curve with rejection regions past +- ???

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(2) Decision Rule:
(a) If obtained value is in the critical region, you “Reject H0”
(b) If obtained value is not in the critical region, you “Fail to Reject H0”
VI. Where and Why Correlations are Used
(A) Prediction
* Example: use SAT scores to predict college GPA (calculate a correlation)
(B) Validity (does the test measure what it claims to measure)
* Example: a new measure of IQ should correlate with other measures of intelligence
(C) Reliability (a test produces stable, consistent measurements)
* Example: a person scoring 112 on an IQ test last week should have a similar score this week
(D) Theory verification
* Example: a theory might predict a relationship between heat and aggression
(this can be verified via a correlation)
VII. Interpreting Correlations
NOTE: There are four things to keep in mind when interpreting a correlation.
(A) Correlation does not necessarily imply causation
(1) If X is correlated with Y, it is possible that (a) X causes Y, (b) Y causes X, or (c) another
variables caused the relationship. A correlation is not an experiment wherein you can
make cause and effect statements.
(2) Example: In U.S. cities the following two variables are significantly correlated: (1) the
number of churches and (2) the number of violent crimes. Do churches cause crime?
No. The real cause of the relationship is a third variable – the population size.
(B) Correlation and restricted range
(1) Issue  You need to be cautious whenever a correlation is computed from scores that do
not represent the full range of possible values.
(2) Example: Relationship of IQ and creativity (but you only sample college students who
probably have a limited IQ range of 110 to 130)
(3) see Figure 2 (on page 6 of lecture)
(4) What to do  Do not generalize the findings beyond the range represented in the sample

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(C) Outliers
(1) Issue  Outliers (extreme scores) can dramatically effect a correlation value
(2) see Figure 14.7 (on page 7 of lecture)
(3) What to do  Look at a scatterplot of your scores to look for outliers
(D) Correlation and the strength of the relationship
(1) A correlation does not imply a proportion
(e.g., r = +.50 does not mean that you can make predictions with 50% accuracy)
(2) Coefficient of Determination
(a) Symbol: r2 (you square the Pearson correlation value – if r = .80, r2 = .64)
(b) Definition: The proportion of variability in one variable that can be determined from
the relationship with the other variable.
(c) Example: if r = .80, then 64% [(.80)2 or .64] of the variability in Y scores can be
predicted from the relationship with X.
VIII. The Point-Biserial Correlation
(A) When is it used?  When you want to assess the relationship between two variables and one of the
variables is continuous (e.g., IQ) and the second variable is dichotomous –
has only 2 values (e.g., college graduate vs. not a college graduate).
(B) Calculation
(1) Step #1  Convert the dichotomous variable so that one category has a value of zero (0)
and the other category has a value of one (1).
(2) Step #2  Compute the regular Pearson correlation using this data
IX. The Spearman Correlation
(A) When is it used?
(1) When you want to measure the relationship between two variables that are measured on an
ordinal scale (rank order)
(2) You want to assess for a “consistent” (monotonic) relationship between two variables not
necessarily a linear relationship
(B) see Figure 14.11 (page 7 of lecture)
** Amount of practice & level of performance

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X. Introduction to Regression
(A) Goal  To identify the straight line that provides the best fit for the data (the “regression line”)
** see Figure 14.13 (on page 8 of lecture) – SAT & College GPA
(B) Purposes of the Line of Best Fit
(1) Make the relationship easier to see
(2) Identifies the center (or central tendency) of the relationship
(3) Allows for prediction
** Example: if you know someone’s SAT score, you can predict their college GPA
(Figure 14.13  SAT of 620 predicts a GPA of 3.25)
(C) Linear Equations
(1) Equation for a line: Y = bX + a (like the Y=mX + b you learned in grade school math)
(2) b = slope of the line
(3) a = y-intercept
(D) How to define the “Line of Best Fit”
(1) Issue  There are multiple lines that can go through a scatterplot. How do we get the one
line that provides the best fit of the data?
(2) Solution  Least Squares Solution
(a) Find the line that has the smallest value for the squared distances from the data points
to the regression line (error).
(b) see Figure 14.15 (on page 8 of lecture)
(3) Formulas for slope and the y-intercept
(a) b = SP / SSX
(b) a = MY – bMX [where MY = mean of Y scores and MX = mean of X scores)
(4) Regression Line
Y = bX + a