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Review of ANOVA and linear regression
Review of simple ANOVA
ANOVAfor comparing means between more than 2 groups
Hypotheses of One-Way ANOVA
All population means are equal i.e., no treatment effect (no variation in means
among groups)
At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are
different (some pairs may be the same)
c3210 μμμμ:H
same the are means population the of all Not:H1
The F-distribution A ratio of variances follows an F-
distribution:
22
220
:
:
withinbetweena
withinbetween
H
H
The F-test tests the hypothesis that two variances are equal. F will be close to 1 if sample variances are equal.
mnwithin
between F ,2
2
~
How to calculate ANOVA’s by hand… Treatment 1 Treatment 2 Treatment 3 Treatment 4
y11 y21 y31 y41
y12 y22 y32 y42
y13 y23 y33 y43
y14 y24 y34 y44
y15 y25 y35 y45
y16 y26 y36 y46
y17 y27 y37 y47
y18 y28 y38 y48
y19 y29 y39 y49
y110 y210 y310 y410
n=10 obs./group
k=4 groups
The group means
10
10
11
1
jjy
y10
10
12
2
jjy
y10
10
13
3
jjy
y 10
10
14
4
jjy
y
The (within) group variances
110
)(10
1
211
j
j yy
110
)(10
1
222
j
j yy
110
)(10
1
233
j
j yy
110
)(10
1
244
j
j yy
Sum of Squares Within (SSW), or Sum of Squares Error (SSE)
The (within) group variances110
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1
211
j
j yy
110
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222
j
j yy
110
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1
233
j
j yy
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4
1
10
1
2)(i j
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+
10
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211 )(
jj yy
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222 )(
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233 )(
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jj yy++
Sum of Squares Within (SSW) (or SSE, for chance error)
Sum of Squares Between (SSB), or Sum of Squares Regression (SSR)
Sum of Squares Between (SSB). Variability of the group means compared to the grand mean (the variability due to the treatment).
Overall mean of all 40 observations (“grand mean”)
40
4
1
10
1
i jijy
y
24
1
)(10
i
i yyx
Total Sum of Squares (SST)
Total sum of squares(TSS).Squared difference of every observation from the overall mean. (numerator of variance of Y!)
4
1
10
1
2)(i j
ij yy
Partitioning of Variance
4
1
10
1
2)(i j
iij yy
4
1
2)(i
i yy
4
1
10
1
2)(i j
ij yy=+
SSW + SSB = TSS
10x
ANOVA Table
Between (k groups)
k-1 SSB(sum of squared deviations of group means from grand mean)
SSB/k-1 Go to
Fk-1,nk-k
chart
Total variation
nk-1 TSS(sum of squared deviations of observations from grand mean)
Source of variation
d.f.
Sum of squares
Mean Sum of Squares
F-statistic p-value
Within(n individuals per
group)
nk-k SSW (sum of squared deviations of observations from their group mean)
s2=SSW/nk-k
knkSSW
kSSB
1
TSS=SSB + SSW
Example
Treatment 1 Treatment 2 Treatment 3 Treatment 4
60 inches 50 48 47
67 52 49 67
42 43 50 54
67 67 55 67
56 67 56 68
62 59 61 65
64 67 61 65
59 64 60 56
72 63 59 60
71 65 64 65
Example
Treatment 1 Treatment 2 Treatment 3 Treatment 4
60 inches 50 48 47
67 52 49 67
42 43 50 54
67 67 55 67
56 67 56 68
62 59 61 65
64 67 61 65
59 64 60 56
72 63 59 60
71 65 64 65
Step 1) calculate the sum of squares between groups:
Mean for group 1 = 62.0
Mean for group 2 = 59.7
Mean for group 3 = 56.3
Mean for group 4 = 61.4
Grand mean= 59.85 SSB = [(62-59.85)2 + (59.7-59.85)2 + (56.3-59.85)2 + (61.4-59.85)2 ] xn per group= 19.65x10 = 196.5
Example
Treatment 1 Treatment 2 Treatment 3 Treatment 4
60 inches 50 48 47
67 52 49 67
42 43 50 54
67 67 55 67
56 67 56 68
62 59 61 65
64 67 61 65
59 64 60 56
72 63 59 60
71 65 64 65
Step 2) calculate the sum of squares within groups:
(60-62) 2+(67-62) 2+ (42-62) 2+ (67-62) 2+ (56-62)
2+ (62-62) 2+ (64-62) 2+ (59-62) 2+ (72-62) 2+ (71-62) 2+ (50-59.7) 2+ (52-59.7) 2+ (43-59.7) 2+67-59.7) 2+ (67-59.7) 2+ (69-59.7) 2…+….(sum of 40 squared deviations) = 2060.6
Step 3) Fill in the ANOVA table
3 196.5 65.5 1.14 .344
36 2060.6 57.2
Source of variation
d.f.
Sum of squares
Mean Sum of Squares
F-statistic
p-value
Between
Within
Total 39 2257.1
Step 3) Fill in the ANOVA table
3 196.5 65.5 1.14 .344
36 2060.6 57.2
Source of variation
d.f.
Sum of squares
Mean Sum of Squares
F-statistic
p-value
Between
Within
Total 39 2257.1
INTERPRETATION of ANOVA:
How much of the variance in height is explained by treatment group?
R2=“Coefficient of Determination” = SSB/TSS = 196.5/2275.1=9%
Coefficient of Determination
SST
SSB
SSESSB
SSBR
2
The amount of variation in the outcome variable (dependent variable) that is explained by the predictor (independent variable).
ANOVA example
S1a, n=25 S2b, n=25 S3c, n=25 P-valued
Calcium (mg) Mean 117.8 158.7 206.5 0.000SDe 62.4 70.5 86.2
Iron (mg) Mean 2.0 2.0 2.0 0.854
SD 0.6 0.6 0.6
Folate (μg) Mean 26.6 38.7 42.6 0.000
SD 13.1 14.5 15.1
Zinc (mg)Mean 1.9 1.5 1.3 0.055
SD 1.0 1.2 0.4a School 1 (most deprived; 40% subsidized lunches).b School 2 (medium deprived; <10% subsidized).c School 3 (least deprived; no subsidization, private school).d ANOVA; significant differences are highlighted in bold (P<0.05).
Table 6. Mean micronutrient intake from the school lunch by school
FROM: Gould R, Russell J, Barker ME. School lunch menus and 11 to 12 year old children's food choice in three secondary schools in England-are the nutritional standards being met? Appetite. 2006 Jan;46(1):86-92.
Answer
Step 1) calculate the sum of squares between groups:
Mean for School 1 = 117.8
Mean for School 2 = 158.7
Mean for School 3 = 206.5
Grand mean: 161
SSB = [(117.8-161)2 + (158.7-161)2 + (206.5-161)2] x25 per group= 98,113
Answer
Step 2) calculate the sum of squares within groups:
S.D. for S1 = 62.4
S.D. for S2 = 70.5
S.D. for S3 = 86.2
Therefore, sum of squares within is:
(24)[ 62.42 + 70.5 2+ 86.22]=391,066
Answer
Step 3) Fill in your ANOVA table
Source of variation
d.f.
Sum of squares
Mean Sum of Squares
F-statistic
p-value
Between 2 98,113 49056 9 <.05
Within 72 391,066 5431
Total 74 489,179
**R2=98113/489179=20%
School explains 20% of the variance in lunchtime calcium intake in these kids.
Beyond one-way ANOVA
Often, you may want to test more than 1 treatment. ANOVA can accommodate more than 1 treatment or factor, so long as they are independent. Again, the variation partitions beautifully!
TSS = SSB1 + SSB2 + SSW
Linear regression review
What is “Linear”?
Remember this: Y=mX+B?
B
m
What’s Slope?
A slope of 2 means that every 1-unit change in X yields a 2-unit change in Y.
Regression equation…
iii xxyE )/(Expected value of y at a given level of x=
Predicted value for an individual…
yi= + *xi + random errori
Follows a normal distribution
Fixed – exactly on the line
Assumptions (or the fine print)
Linear regression assumes that… 1. The relationship between X and Y is linear 2. Y is distributed normally at each value of X 3. The variance of Y at every value of X is the
same (homogeneity of variances) 4. The observations are independent**
**When we talk about repeated measures starting next week, we will violate this assumption and hence need more sophisticated regression models!
The standard error of Y given X is the average variability around the regression line at any given value of X. It is assumed to be equal at all values of X.
Sy/x
Sy/x
Sy/x
Sy/x
Sy/x
Sy/x
C A
B
A
yi
x
y
yi
C
B
*Least squares estimation gave us the line (β) that minimized C2
ii xy
y
A2 B2 C2
SStotal
Total squared distance of observations from naïve mean of y Total variation
SSreg Distance from regression line to naïve mean of y
Variability due to x (regression)
SSresidual
Variance around the regression line
Additional variability not explained by x—what least squares method aims to minimize
n
iii
n
i
n
iii yyyyyy
1
2
1 1
22 )ˆ()ˆ()(
Regression Picture
R2=SSreg/SStotal
Recall example: cognitive function and vitamin D
Hypothetical data loosely based on [1]; cross-sectional study of 100 middle-aged and older European men. Cognitive function is measured by the
Digit Symbol Substitution Test (DSST).
1. Lee DM, Tajar A, Ulubaev A, et al. Association between 25-hydroxyvitamin D levels and cognitive performance in middle-aged and older European men. J Neurol Neurosurg Psychiatry. 2009 Jul;80(7):722-9.
Distribution of vitamin D
Mean= 63 nmol/L
Standard deviation = 33 nmol/L
Distribution of DSST
Normally distributed
Mean = 28 points
Standard deviation = 10 points
Four hypothetical datasets
I generated four hypothetical datasets, with increasing TRUE slopes (between vit D and DSST): 0 0.5 points per 10 nmol/L 1.0 points per 10 nmol/L 1.5 points per 10 nmol/L
Dataset 1: no relationship
Dataset 2: weak relationship
Dataset 3: weak to moderate relationship
Dataset 4: moderate relationship
The “Best fit” line
Regression equation:
E(Yi) = 28 + 0*vit Di (in 10 nmol/L)
The “Best fit” line
Note how the line is a little deceptive; it draws your eye, making the relationship appear stronger than it really is!
Regression equation:
E(Yi) = 26 + 0.5*vit Di (in 10 nmol/L)
The “Best fit” line
Regression equation:
E(Yi) = 22 + 1.0*vit Di (in 10 nmol/L)
The “Best fit” line
Regression equation:
E(Yi) = 20 + 1.5*vit Di (in 10 nmol/L)
Note: all the lines go through the point (63, 28)!
Significance testing…Slope
Distribution of slope ~ Tn-2(β,s.e.( ))
H0: β1 = 0 (no linear relationship)
H1: β1 0 (linear relationship does exist)
)ˆ.(.
0ˆ
es
Tn-2=
Example: dataset 4
Standard error (beta) = 0.03 T98 = 0.15/0.03 = 5, p<.0001
95% Confidence interval = 0.09 to 0.21
Multiple linear regression…
What if age is a confounder here? Older men have lower vitamin D Older men have poorer cognition
“Adjust” for age by putting age in the model: DSST score = intercept +
slope1xvitamin D + slope2 xage
2 predictors: age and vit D…
Different 3D view…
Fit a plane rather than a line…
On the plane, the slope for vitamin D is the same at every age; thus, the slope for vitamin D represents the effect of vitamin D when age is held constant.
Equation of the “Best fit” plane… DSST score = 53 + 0.0039xvitamin D
(in 10 nmol/L) - 0.46 xage (in years)
P-value for vitamin D >>.05 P-value for age <.0001
Thus, relationship with vitamin D was due to confounding by age!
Multiple Linear Regression More than one predictor…
E(y)= + 1*X + 2 *W + 3 *Z…
Each regression coefficient is the amount of change in the outcome variable that would be expected per one-unit change of the predictor, if all other variables in the model were held constant.
Functions of multivariate analysis:
Control for confounders Test for interactions between predictors
(effect modification) Improve predictions
ANOVA is linear regression!
Divide vitamin D into three groups: Deficient (<25 nmol/L) Insufficient (>=25 and <50 nmol/L) Sufficient (>=50 nmol/L), reference group
DSST= (=value for sufficient) + insufficient*(1 if insufficient) + 2 *(1 if deficient)
This is called “dummy coding”—where multiple binary variables are created to represent being in each category (or not) of a categorical variable
The picture…
Sufficient vs. Insufficient
Sufficient vs. Deficient
Results… Parameter Estimates
Parameter Standard Variable DF Estimate Error t Value Pr > |t|
Intercept 1 40.07407 1.47817 27.11 <.0001 deficient 1 -9.87407 3.73950 -2.64 0.0096 insufficient 1 -6.87963 2.33719 -2.94 0.0041
Interpretation: The deficient group has a mean DSST 9.87
points lower than the reference (sufficient) group.
The insufficient group has a mean DSST 6.87 points lower than the reference (sufficient) group.