Evidence based practitioners ANOVA

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Analysis of Variance OT 667
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ANOVA is a powerful statistical tool developed in
the 1950s by Sir Ronald Fisher
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What is analysis of variance? (ANOVA)

• A statistical procedure for comparing three or
  more conditions or groups to see if the
  difference among group variances are greater than
  would be expected by chance alone.
• The critical value is calculated on the F
  statistic, a ratio of between groups effects to
  within group variability.

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Different Forms of ANOVA

• One-way ANOVA
• Repeated measures ANOVA
• Two and three -way ANOVA (factorial designs)
• Mixed designs, where one factor is repeated but
  another is held constant (factorial designs)

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One Way ANOVA Group A Group
B Group C Scores X1
X1 X1
X2 X2 X2
. . .
. .
.
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One-way ANOVA

• Grand mean
• Sum of squares
• Partitioning the variability - between groups
  (treatment effect) and within groups (error term)
• Degrees of freedom
• Mean Squares
• F statistic

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Sums of squares

• ANOVA uses sums of squares to analyze
  differences between groups __
• SS S(X - X )2
• The larger the sum of squares, the greater the
  variability in a data set

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Partitioning the Variability

• Calculate sums of squares for the following
• total sum of squares for all subjects
• sums of squares for between groups
• (treatment variance)
• sums of squares for within groups
• (error variance)

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Degrees of freedom in ANOVA

• Degrees of freedom are stated for the various
  calculations, such as for total subjects, for
  between groups, and for within groups. They are
  used to help calculate the various F ratios

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Mean Squares

• Sums of squares are converted into a variance
  estimate or mean square
• Each sum of squares is divided by its respective
  degrees of freedom to get the mean square

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F statistic (after Ronald Fisher who developed
ANOVA)

• The F statistic is a ratio between calculated by
  dividing the mean square between groups
  (treatment variance) by the mean square within
  groups (error variance)
• F MSb
• MSe

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Post-hoc tests

• ANOVA only tells you there is a difference
  between groups, not which group shows greatest
  change
• Use post-hoc tests to tell you which group is
  different
• Newman- Keuls, Tukeys honestly significant
  difference, Scheffes test

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Factorial ANOVA (2 x 2) Group A Group
BVariable A s1
s1 s2 s2 s3
s3 Variable B
s1 s1 s2
s2 s3 s3
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Case-Smith Study (2002)

• Compared with a control group of students with
  poor handwriting who do not receive occupational
  therapy, will students with poor handwriting who
  receive occupational therapy services make
  greater improvements in visual-motor
  skill,visual-perception skill, dexterity, in-hand
  manipulation skills, legibility and handwriting
  speed?

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Independent variables - group (one gets OT, one
does not), time (pre and post measures)Dependent
variables - visual-motor skill,
visual-perception skill, dexterity, in-hand
manipulation skills, legibility, and handwriting
speed
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What are the effects of variable A, independent
of variable B? (time using pre and post
measures)What are the effects of variable B
independent of variable A?(group)What is the
joint effect or interaction of variables A and
B?(time by group)
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Calculations for Higher order ANOVAs

• Basic methods are the same, i.e. use of grand
  mean, sum of squares to calculate mean squares
• Higher order ANOVAs allow more comparisons
  between and within as independent variables and
  levels of variables are added

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Intervention
Control Group (25)
Group     DTVP Position in space sum of
all scores sum of all scores Copying sum of
all scores sum of all scores FG
perception sum of all scores sum of all
scores   BOTMP V-M control sum of all scores
sum of all scores UL speed/dexterity sum of
all scores sum of all scores


Nine hole peg
test sum of all scores sum of all
scores   Step One 1. Calculating a grand
mean add ALL scores together and calculate a
mean Step Two 2. Subtract squared
deviation scores for ALL subjects from the grand
mean Step Three 3. Calculate
difference in sum of squares between the two
groups (this is the between groups
variance) Step Four 4. Subtract sum of
squares for each group from the grand mean
(this is the
within groups or error variance)      
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Main effects - The effect of each independent
variable judged separately of each other
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IG
CG
BOTMP VM
BOTMP S/D
The difference between group scores on the BOTMP
visual motor control subtest and the upper limb
speed and dexterity subtest is known as a main
effect.
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Pre
Post
DVPT copying
DVPT FG
DVPT P in S
The difference between between pre and post
scores (time) on the DVPT copying, figure-ground
and position in space subtests is another main
effect.
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All Main Effects for Case-Smith Study

• 8 main effects for time
• 8 main effects for group

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Interaction effects are seen across cells of each
independent variable when the effects of one
variable are not constant across the second
variable
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Interactions in Case-Smith studyEffect of Group
and Time on various test scores
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Significant Outcomes in Case-Smith Study

• Three significant main effects reported
• Main effect for group (IG) on in-hand
  manipulation
• Main effect for group (IG) on visual-motor
  control
• Main effect for group (IG) on percentage of
  legible letters

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No interaction effects reported (inconsistencies
between table and text)
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In summary...

• ANOVA is a very powerful statistical test that
  enables the researcher to compare the effects of
  multiple independent variables on a single
  dependent variable