


Sherril M. Stone, Ph.D. 

Department of Family Medicine 

OSUCollege of Osteopathic Medicine 




SS (Sum of squares) Ð sum of squared deviations
 SX^{2}
 (SX)^{2} 

N 

MS (Mean square) Ð SS/df  this is estimate of population
variance 

SX_{tot}
(G  Grand mean) Ð mean of all scores 

SX_{tot}
(G  Grand total) Ð sum of all
scores (XÕs) added together 

T or t (Treatment groups) Ð T_{1}, T_{2},
T_{n}, each treatment group 

X (scores) Ð scores in each group 

K Ð the number of levels of IV 

F test Ð the ANOVA 

n  the number of scores (XÕs) in each treatment 

N Ð
the total number of scores (XÕs) for all treatments 

Between groups Ð variability (difference) between
treatment groups 

Within groups Ð variability in a single group  AKA the
error 







Similar to ttest Ð except IV has 2 or more
levels 

OneWay 

1 IV analyzed and 1 DV measured 

H_{0}: population means are all equal (H_{0}:
µ_{1} = µ_{2} = µ_{3}) 

H_{1}: population means are NOT equal (µ_{1}
µ_{2} µ_{3}) 

Treatment Ð may have more than 2 levels 

Hypothesis testing 

H_{0} (null) and H_{1} (alternate) hypotheses are
stated (Assume H_{0} is true) 

Sampling distribution Ð 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 Ð treatment/between groups variation to
within group variation 

B  between groups Ð do groups differ due to
treatment 

W or E within (error) Ð unbiased estimate of
variability in same group 

B > W reject Ho (treatment had effect) 

Small variability (F=1.00) Ð keep H_{0},
large var (F > 1.00) Ð reject H_{0} 





Assumptions of ANOVA 

Normality 

Homogeneity of variance 

Random assignment 

Effect Size 

r^{2} = SS_{B} f = .10 Small 

SS_{tot} f
= .25 Medium 

f = .40 Large 




Oneway (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) 






A measure of variability 

SS_{tot}=SS_{btwn}+SS_{wthn} IS SS_{tot}=SS_{treat}+SS_{error} 

All SS are 0 or positive number 

SS_{tot }(SS_{tot }= SS_{btwn
}+ SS_{wthn}) 

SX^{2}_{tot} 
(SX_{tot})^{2} 

N 



SS_{btwn} 

(SX_{t})^{2} (SX_{tot})^{2} 

S N N 




Each SS has a df 

df = n Ð 1 

df_{tot} = N Ð 1 (df_{tot} = df_{w}
+ df_{b}) 



df_{b} = K Ð 1 (K = number of groups) 



df_{w} = N Ð K (K = number of groups) 





t^{2}
= F (with 1 df) 

t^{2}
(15 df) = F (1, 15 df) 

ttest is based upon mean differences 

Ftest 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 







MS = variance (variance is the mean of the
squared deviations) 

MS = S^{2} = SS/df 

MS_{B} = SS_{b}/df_{B} 

MSw = SS_{W}/df_{W} 

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 





Calculate descriptives 

Calculate SS 

SS_{tot} 

SS_{B} 

SS_{W} 

Calculate df 

df_{tot} 

df_{B} 

df_{W} 








Source SS df MS F 

Between 

Within 

Total 



Always label your source with the variable names 




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 




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 




Source SS df MS F 

Treatment 

Error 

Total 






Participants provide more than 1 score 

Participants are included in all IV levels 

SXs =
sum of each subjectÕs scores 

Each participant serves as own control 

Design requires fewer participants 

Results have higher validity and reliability 




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 






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 








Source SS df MS F 

Women 

Treatment 

Error 

Total 




Betweensubjects, Withinsubjects, B x W, RM 

Factor = IV (2 or more levels) 

Factorial = 2 or more IVs (i.e. 1way 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) 





SS_{main} 

SS_{A} = (SX_{A1})^{2} (SX _{A2})^{2} (SX _{A3})^{2} (SX_{tot})^{2} 

N_{A}
N _{A2} N_{A3}
N_{tot} 



SS_{B} = (SX_{B1})^{2} (SX _{B2})^{2} (SX_{tot})^{2} 

N_{B}
N _{B2} N_{tot} 





SS_{AB }= N_{cell} (A_{1}B_{1}  A_{1}
 B_{1} + tot)^{2} + (A_{2}B_{1}  A_{2}
 B_{1} + tot)^{2} 

+ (A_{3}B_{1}  A_{3}  B_{1} + tot)^{2}
+ (A_{1}B_{2}  A_{1}  B_{2} + tot)^{2} 

+ (A_{2}B_{2}  A_{2}  B_{2} + tot)^{2}
+ (A_{3}B_{2}  A_{3}  B_{2} + tot)^{2} 




Six groups of obese (100120 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? 




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 




Source SS df MS F 

Treatment 

Exercise 

Diet 

E x D 

Error 

Total 






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? 




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 




Source SS df MS F 

Treatment 

Drug 

Error 

Total 







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 




Source SS df MS F 

Treatment 

Drug 

Diet 

D x D 

Error 

Total 




