One Way WithinOne-Way Within-Groups ANOVAGroups ANOVAPSYC 381 – StatisticsArlo Clark-Foos, Ph.D.

Comparing DesignsComparing Designs• Pros of Between

• No order or carryover effects

Between-Groups Design

• Pros of Within• More costly• Higher variabilityg y

between groupsthan within(i e individual differences)(i.e., individual differences)

Used A Lot in Market Research!Used A Lot in Market Research!Taste Tests

Odor Tests

Within Groups Designs What about order/carryover

effects?Within Groups Designs• All participants experience all conditions/treatments/levels

effects?

Counterbalance!

CounterbalancingCounterbalancing• Minimization of order effects by varying the order of presentation of

different levels of the independent variable from one participant (ordifferent levels of the independent variable from one participant (or group) to the next.

• An example of a counterbalanced within-subjects design with 3 conditions:

3 Conditions/Levels of IV3! = 3 x 2 x 1 = 6 orders

What if your IV had more levels?

Example: IV = Year in School (4 levels)4 x 3 x 2 x 1 = 24 orders!!

You need a Latin Square Design!

Latin Square DesignLatin Square DesignA technique to control for order effects without having all possible orders A limited set of orders isorders. A limited set of orders is constructed to ensure that (1) each condition appears at each ordinal position and (2) each condition precedes and follows each condition one time.

Within Groups ANOVAWithin Groups ANOVA• New Terminology

• SSSubjects

• dfSubjects

• n• n

• A few new formulas (along with the old ones)

An Example with Beer!An Example with…Beer!• Do you have a love of lagers? • A journalist (Fallows, 1999) wanted to know if self-proclaimed

beer snobs would be able to distinguish between three classes/qualties of beers. There are over 50+ styles of beer, many of which are not available in lower quality versions so he chose lagers because of their widespread availability. The results below are their taste ratings for each beer.

• Cheap Beers• e.g.,

• Mid-Range Beers• e.g., Budweise

• High-End Beers• e.g.,

One-Way Within Groups ANOVA: Beer Taste Testing• Six Steps to Hypothesis Testing…1. Identify the populations,

1. People who drink cheap beer2. People who drink mid-range beerp g3. People who drink high-range beer

• Distribution F distribution (>2 groups)Distribution, F distribution (>2 groups)• One-Way Within-Groups ANOVA

• Assumptions• Assumptions1. Participants not selected randomly, careful generalizing2. Data do not appear skewed3 H d ti it [(l t i ) ≤ (2 ll t i )]3. Homoscedasticity [(largest variance) ≤ (2 x smallest variance)]4. Are there order effects? Not counterbalanced…

One-Way Within Groups ANOVA: Beer Taste Testing2. State null and research hypotheses

Null: People who drink cheap, mid-range, and high-end beer rate their beers the same on averagerate their beers the same, on average.

Research: People who drink cheap, mid-range, and high-end beer do not rate their beers the same, on average.

One-Way Within Groups ANOVA: Beer Taste Testing3. Determine characteristics of comparison distribution

2131 =−=−= GroupsBetween Ndf 4151 =−=−= ndfSubjects

( )( ) ( )( ) 842 === SubjectsBetweenWithin dfdfdf

GroupsBetweenf fSubjects

14842 =++=++= WithinSubjectsBetweenTotal dfdfdfdf

or141151 =−=−= TotalTotal Ndf

One-Way Within Groups ANOVA: Beer Taste Testing4. Determine the critical values or cutoffs (p = 05).

2=BetweendfBetweenf

8=Withindf

46.4=CriticalF

One-Way Within Groups ANOVA: Beer Taste Testing5. Calculate the test statistic

( )2GMXSSTotal −=

One-Way Within Groups ANOVA: Beer Taste Testing5. Calculate the test statistic

( )2GMMSSBetween −=

One-Way Within Groups ANOVA: Beer Taste Testing5. Calculate the test statistic ( )2GMMSS tP ti iS bj t −Σ= ( )GMMSS tParticipanSubjects Σ

One-Way Within Groups ANOVA: Beer Taste Testing5. Calculate the test statistic

SubjectsBetweenTotalWithin SSSSSSSS −−=

73872913510927322117859295 = 738.729135.1092732.2117859.295 −−=

One-Way Within Groups ANOVA: Beer Taste Testing6. Make a decision

People who dink cheap, mid-range, and high-end beers do not p p, g , grate their beers the same, on average, F(2, 14) = 14.77, p < .05

( ) ( )46.477.14)14,2( =>= CriticalFF

Effect SizeEffect Size2 SS 2 1092 1352

( )Between

Total Subjects

SSRSS SS

=−

2 1092.135 .787(2117.732 729.738)

R = =−

SummarySummary• Pros & Cons of Within & Between Subjects Designs

• Order Effects• Counterbalancing & Latin Square

• New Sums of Squares and Degrees of Freedom (Subjects)• New Source of Variability

• Effect Size