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

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1

244

j

j yy

Sum of Squares Within (SSW), or Sum of Squares Error (SSE)

The (within) group variances110

)(10

1

211

j

j yy

110

)(10

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222

j

j yy

110

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233

j

j yy

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j

j yy

4

1

10

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2)(i j

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10

1

211 )(

jj yy

10

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222 )(

jj yy

10

3

233 )(

jj yy

10

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244 )(

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

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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.