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Basic Concepts of One-way Analysis of Variance (ANOVA). Sporiš Goran, PhD. http://kif.hr/predmet/mki http://www.science4performance.com/. Overview. What is ANOVA? When is it useful? How does it work? Some Examples Limitations Conclusions. Definitions.
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Basic Concepts of One-wayAnalysis of Variance (ANOVA) Sporiš Goran, PhD. http://kif.hr/predmet/mki http://www.science4performance.com/
Overview • What is ANOVA? • When is it useful? • How does it work? • Some Examples • Limitations • Conclusions
Definitions • ANOVA: analysis of variation in an experimental outcome and especially of a statistical variance in order to determine the contributions of given factors or variables to the variance. • Remember: Variance: the square of the standard deviation Remember: RA Fischer, 1919-Evolutionary Biology
Introduction • Any data set has variability • Variability exists within groups… • and between groups • Question that ANOVA allows us to answer : Is this variability significant, or merely by chance?
The difference between variation within a group and variation between groups may help us determine this. If both are equal it is likely that it is due to chance and not significant. • H0: Variability w/i groups = variability b/t groups, this means that 1 = n • Ha: Variability w/i groups does not = variability b/t groups, or, 1 n
Assumptions • Normal distribution • Variances of dependent variable are equal in all populations • Random samples; independent scores
One-Way ANOVA • One factor (manipulated variable) • One response variable • Two or more groups to compare
Usefulness • Similar to t-test • More versatile than t-test • Compare one parameter (response variable) between two or more groups
For instance, ANOVA Could be Used to: • Compare heights of plants with and without galls • Compare birth weights of deer in different geographical regions • Compare responses of patients to real medication vs. placebo • Compare attention spans of undergraduate students in different programs at PC.
Why Not Just Use t-tests? • Tedious when many groups are present • Using all data increases stability • Large number of comparisons some may appear significant by chance
Remember that… • Standard deviation (s) n s = √[(Σ (xi– X)2)/(n-1)] i = 1 • In this case: Degrees of freedom (df) df = Number of observations or groups - 1
Notation • k = # of groups • n = # observations in each group • xij = one observation in group i • Y = mean over all groups • Yi = mean for group i • SS = Sum of Squares • MS = Mean of Squares • λ = Between MS/Within MS
FYI this is how SS Values are calculated k ni • Total SS = Σ Σ (xij–)2 = SStot i=1 j=1 k ni • Within SS = Σ Σ (xij–i)2 = SSw i=1 j=1 k ni • Between SS = Σ Σ (i–)2 = SSbet i=1 j=1
and • SStot = SSw + SSbet
Calculating MS Values • MS = SS/df • For between groups, df = k-1 • For within groups, df= n-k
F-Ratio = MSBet/MSw • If: • The ratio of Between-Groups MS: Within-Groups MS is LARGE reject H0 there is a difference between groups • The ratio of Between-Groups MS: Within-Groups MS is SMALLdo not reject H0 there is no difference between groups
p-values • Use table in stats book to determine p • Use df for numerator and denominator • Choose level of significance • If F > critical value, reject the null hypothesis (for one-tail test)
Example 1, pp. 400 of your handout • Three groups: • Middle class sample • Persons on welfare • Lower-middle class sample • Question: Are attitudes toward welfare payments the same?
and From the table with = 0.05 and df = 2 and 24, we see that if F > 3.40 we can reject Ho. This is what we would conclude in this case.
Example 2 • Bat cave gates: • Group 1 = No gate (NG) • Group 2 = Straight entrance gate (SE) • Group 3 = Angled entrance gate (AE) • Group 4 = Straight dark zone gate (SD) • Group 5 = Angled dark zone gate (AD) • Question: Is variation in bat flight speed greater within or between groups? Or Ho = no difference significant difference in means.
Group #, i Gate Type Mean FS (m/s) sd FS (m/s) ni 1 NG 5.6 0.93 150 2 SE 3.8 1.05 150 3 AE 4.7 0.97 150 4 SD 4.2 1.23 137 5 AD 5.1 1.03 143 Example 2 (cont’d) Hypothetical data for bat flight speed with various gate arrangements. FS= Flight speed; sd = standard deviation
Group #, i Gate Type Mean FS (m/s) sd FS (m/s) ni 1 NG 5.6 0.93 150 2 SE 3.8 1.05 150 3 AE 4.7 0.97 150 4 SD 4.2 1.23 137 5 AD 5.1 1.03 143 Example 2 SSbet Between SS= 300
Group #, i Gate Type Mean FS (m/s) sd FS (m/s) ni 1 NG 5.6 0.93 150 2 SE 3.8 1.05 150 3 AE 4.7 0.97 150 4 SD 4.2 1.23 137 5 AD 5.1 1.03 143 Example 2 SSw Within SS = 790
Example 2 (cont’d) • Between MS = 300/4 = 75 • Within MS = 790/(730-5) = 1.09 • F Ratio = 75/1.09 = 68.8 • See Table find p-value based on df= 4, • Since F>value found on the table we reject Ho.
What ANOVA Cannot Do • Tell which groups are different • Post-hoc test of mean differences required • Compare multiple parameters for multiple groups (so it cannot be used for multiple response variables)
Some Variations • Two-Way, Three-Way, etc. ANOVA (will talk about this next class) • 2+ factors • MANOVA (Multiple analysis of variance) • multiple response variables
Summary • ANOVA: • allows us to know if variability in a data set is between groups or merely within groups • is more versatile than t-test • can compare multiple groups at once • cannot process multiple response variables • does not indicate which groups are different
Now, let’s go to our SPSS manual • Perform the sample problem on the effects of attachment styles on the psychology of sleep with the data set from the NAAGE site called Delta Sleep. • Pay attention to the procedure and the post-hoc tests to determine which groups are significantly different. Perform the Tukey Test at a 5% significance level. • Look at your output and interpret your results. • Tell me when you are done.
When you are done with this, • Do practice exercises 1, 4, 6, 7 and 12 from the handout in SPSS. • Create the data sets. • Run the one-way ANOVAS and interpret your results.