Notes

Outline

Statistics 8 Analysis of Variance ANOVA Sherril M. Stone, Ph.D. Department of Family Medicine OSU-College of Osteopathic Medicine

Basic Terms SS (Sum of squares) – sum of squared deviations - SX2 - (SX)2                                                     N MS (Mean square) – SS/df   - this is estimate of population variance SXtot (G - Grand mean) – mean of all scores SXtot (G - Grand total)  – sum of all scores (X’s) added together T or t (Treatment groups) – T1, T2, Tn, each treatment group X (scores) – scores in each group K – the number of levels of IV F test – the ANOVA n - the number of scores (X’s) in each treatment  N – the total number of scores (X’s) for all treatments  Between groups – variability (difference) between treatment groups  Within groups – variability in a single group - AKA the error

ANOVA Basics Similar to t-test – except IV has 2 or more levels One-Way 1 IV analyzed and 1 DV measured H0: population means are all equal (H0: µ1 = µ2 = µ3) H1: population means are NOT equal (µ1 ≠ µ2 ≠ µ3) Treatment – may have more than 2 levels Hypothesis testing H0  (null) and H1 (alternate) hypotheses are stated (Assume H0 is true) Sampling distribution – t or F Gather data and calculate F value Compare calculated F to table value and Interpret results F distribution F is a sampling distribution (just as t distribution is) Ratio – treatment/between groups variation to within group variation B - between groups – do groups differ due to treatment W or E within (error) – unbiased estimate of variability in same group B > W reject Ho (treatment had effect) Small variability (F=1.00) – keep H0, large var (F > 1.00) – reject H0

ANOVA Basics Assumptions of ANOVA Normality Homogeneity of variance Random assignment Effect Size r2 = SSB f = .10 Small      SStot f = .25 Medium f = .40 Large

Types of ANOVAs One-way (1 DV, 1 IV with 2 or more levels) Repeated Measures Factorial (1DV, 2 or more IVs with 2 or more levels) MANOVA (2 or more DVs, 2 or more IVs with various number of levels) ANCOVA (ANOVA with correlation included) MANCOVA (MANOVA with correlation included)

Sum of Squares A measure of variability SStot=SSbtwn+SSwthn  IS  SStot=SStreat+SSerror All SS are 0 or positive number SStot (SStot = SSbtwn + SSwthn) SX2tot    -   (SXtot)2          N SSbtwn (SXt)2 (SXtot)2    S    N     N

Degrees of Freedom Each SS has a df df = n – 1 dftot = N – 1  (dftot = dfw + dfb) dfb = K – 1 (K = number of groups) dfw = N – K (K = number of groups)

t and F Similarities  t2 = F (with 1 df)  t2 (15 df) = F (1, 15 df) t-test is based upon mean differences F-test is based upon variance differences t =  calculated difference between sample means                             chance difference (error) F =     variance between sample means    chance variance  (error) Numerator differences btwn groups due to treatment Denominator differences btwn groups due to chance/ error

Variance MS = variance (variance is the mean of the squared deviations) MS = S2 = SS/df MSB = SSb/dfB MSw = SSW/dfW Two possible sources of variance exist Treatment Variance - btwn group differs due to treatment Possible sources of treatment variance Treatment effect Individual differences Experimenter error Error Variance - wthn treatment group participants receive same manipulation.  If their scores differ it is due to error Possible sources of error variance Individual differences Experimenter error

Calculating the F Calculate descriptives Calculate SS SStot SSB SSW Calculate df dftot dfB dfW

ANOVA Summary Table Source SS df MS F Between Within Total Always label your source with the variable names

Example 1 One-way ANOVA Three groups of women volunteer to have a mammogram at different times. The raw score provided is the degree of pain (1=none to 10=extreme) they experienced during the test. Does the testing time affect the degree of pain experienced? Testing Time        9 AM       1 PM       4 PM 9 2 4 6 1 4 6 4 2 8 3 3 7 2 3

Steps to analyze Example 1 Calculate descriptives Calculate SS Calculate df Calculate MS Calculate F Compare F to Table value Prepare ANOVA Summary Table Interpret the results and explain findings

ANOVA Summary Table Source      SS df MS F Treatment Error Total

Repeated Measures ANOVA Participants provide more than 1 score Participants are included in all IV levels SXs = sum of each subject’s scores Each participant serves as own control Design requires fewer participants Results have higher validity and reliability

Example 2 Repeated Measures ANOVA Five women volunteer to have a mammogram at different times (over 3 days). The raw score provided is the degree of pain (1=none to 10=extreme) they experienced during the test. Does the testing time affect the degree of pain experienced? Testing Time        9 AM       1 PM       4 PM 9 2 4 6 1 4 6 4 2 8 3 3 7 2 3

Steps to analyze Example 2 Calculate descriptives Calculate SS Calculate df Calculate MS Calculate F Compare F to Table value Prepare ANOVA Summary Table Interpret the results and explain findings

ANOVA Summary Table Source      SS df MS F Women Treatment Error Total

Factorial ANOVA Between-subjects, Within-subjects, B x W, RM Factor = IV (2 or more levels) Factorial = 2 or more IVs (i.e. 1-way ANOVAs) Main Effect = each IV Interaction Effect = interaction of IVs Cells = different participants exposed to different treatments (IV effects) Column = IV (A) Row = IV (B)

Factorial ANOVA

Factorial SS Formulas SSmain       SSA    = (SXA1)2        (SX A2)2 (SX A3)2          (SXtot)2            NA                  N A2   NA3                Ntot        SSB   = (SXB1)2        (SX B2)2 (SXtot)2            NB                  N B2 Ntot SSAB =  Ncell (A1B1 - A1 - B1 + tot)2 + (A2B1 - A2 - B1 + tot)2    + (A3B1 - A3 - B1 + tot)2 + (A1B2 - A1 - B2 + tot)2    + (A2B2 - A2 - B2 + tot)2 + (A3B2 - A3 - B2 + tot)2

Example 3 Between-Subjects ANOVA Six groups of obese (100-120 lbs overweight) children volunteer to participate in a weight loss study. The raw score is number of lbs lost during the 6 month study. Does type of diet and exercise affect amount of weight loss?

Steps to analyze Example 3 Calculate descriptives Calculate SS Calculate df Calculate MS Calculate F Compare F to Table value Prepare ANOVA Summary Table Interpret the results and explain findings

ANOVA Summary Table Source      SS df MS F Treatment    Exercise    Diet    E x D Error Total

Example 4 Totally Within-Subjects ANOVA Five men with very high cholesterol volunteer to participate in a 6 month study. After each month they do not adhere to any regimen to allow their normal count to stabilize Does drug type affect cholesterol level?

Steps to analyze Example 4 Calculate descriptives Calculate SS Calculate df Calculate MS Calculate F Compare F to Table value Prepare ANOVA Summary Table Interpret the results and explain findings

ANOVA Summary Table Source      SS df MS F Treatment    Drug Error Total

Example 5 Mixed Between x Within ANOVA (3 x 2)

Steps to analyze Example 5 Calculate descriptives Calculate SS Calculate df Calculate MS Calculate F Compare F to Table value Prepare ANOVA Summary Table Interpret the results and explain findings

ANOVA Summary Table Source      SS df MS F Treatment    Drug    Diet    D x D Error Total

Graphing Results

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