ANCOVA

A hybrid of regression and analysis of variance

Analysis of covariance

• It is an analysis of variance performed on residuals from the regression of the response variable on the covariate

Analysis of covariance

ijiijiiij XXAY )(

ijiijCiij XXAY )(

Plotting ANCOVAs

• The ANCOVA plot should use the continuous covariate variable plotted on the x-axis, and the Y variable plotted on the y-axis. Each point represents an independent replicate, and different symbols or colors should be used for each treatment group.

Plotting results A B C

D E F

Match???

1. Treatment significant, covariate and interaction term non-significant (C)

2. Treatment and covariate significant, interaction term non-significant (D)

3. Interaction term significant, everything else non-significant (E)

4. Covariate not significant, treatment, and interaction significant (F)

5. Covariate significant, treatment and interaction non-significant (B)

6. No term significant (A)

Dangerous data!!

0

10

20

30

40

50

60

0 2 4 6 8 10 12

T1

T2

T3

An important thing…

• In Analysis of Covariance order matters

• This model:

model <- lm(Y ~ X*Group)

• is not the same as this one

model <- lm(Y ~ Group*X)

Membrane potential (in millivolts)

www.ebme.co.uk/arts/aps/pic1a.gif

'Action potential' is the name given to the electrical nerve impulse waveform that is generated by the neuron (nerve cell). The shape of an action potential can be seen using an amplifier circuit (voltage clamp) as shown in the diagram below, which measures the flow of ions using two electrodes inserted into the nerve fibre.

Membrane potential (in millivolts)

• Yamauchi and Kimizuka (1971) measured membrane potential for 4 different cation systems as a function of the logarithm of the activity ratio of various electrolytes are various concentrations. We wish to test whether the mean membrane potential “Y” is different for these

systems

Data

a= 4 groups (cation systems)

Ca-Li Ca-Na Ca-K Sr-Na

Y X Y X Y X Y X

-2.4 -0.31 -7.0 -1.18 -10.8 -1.79 -5.4 -1.83

6.3 0.17 2.1 -0.65 -2.8 -1.21 3.0 -1.25

15.8 0.58 17.8 0.10 14.2 -0.35 20.7 -0.41

20.5 0.81 27.3 0.50 25.5 0.08 30.5 0.05

32.0 0.67 35.7 0.49 39.9 0.43

41.2 0.65 45.0 0.59

N 4 5 6 6

10.05 0.312 14.44 -0.112 17.17 -0.355 22.28 -0.403X

Membrane potential for four different cation systems

-40

-30

-20

-10

0

10

20

30

40

50

60

-2 -1.5 -1 -0.5 0 0.5 1

log activity ratio

me

an

me

mb

ran

e p

ote

ntia

l (in

mV

) Ca-Li

Ca-Na

Ca-K

Sr-Na

Linear (Sr-Na)

Linear (Ca-Na)

Linear (Ca-K)

Linear (Ca-Li)

Component Ca-Li Ca-Na Ca-K Sr-Na Pooled Within

(sum)

311.33 1096.97 2180.13 2034.63 5623.06

0.727 2.461 4.703 4.639 12.53

15.02 51.85 100.63 96.80 264.30

20.66 21.07 21.39 20.87

22 )(1

i

n

i YYy

For each group compute the following:

22 )(1

i

n

i XXx

))((1

YYXXxy ii

n

i

1b 09.21withinb

Component Ca-Li Ca-Na Ca-K Sr-Na Pooled(sum)

310.43 1092.43 2152.84 2020.02 5575.73

0.898 4.54 27.29 14.60 47.33

22 )ˆ(ˆ1

i

n

i YYy

For each group compute the following:

2)ˆ(1

i

n

i YY

93.557453.12

)30.274(

)(

)))(((

ˆ2

2

2

2

1

1

i

n

i

ii

n

i

within

XX

YYXX

y 13.4893.557406.56232_ withinYXd

33.4773.557506.56232 YXd

79.033.4713.48'_ sbamongSS

46.242)()(1

YYXXnxy i

n a

iitotal

85.2130.26445.2421

within

n

totalamong xyxyxy

We obtain amongxy

71.1816006.14

)57.242(72.6013

)( 2

2

222

_

1

total

totaln

totaltotalYX x

xyyd

We calculate unexplained sums of squares for these two levels of variation:

309.6748.1

)84.21(65.390

)( 2

2

222

_

1

among

amongn

amongamongYX x

xyyd

111

65.39006.562372.6013222n

within

n

total

n

among yyy

6.1768129.4871.18162_

22)( withinYXYXtotaladjYX ddd

We test the null hypothesis that there are no differences among sample means when these are adjusted for a common and a common regression line:

Y X

53.5983

56.1768

1_

2)(

_

a

dsquareMean adjYX

meansadjusted

008.316

1.48

1_

2)(

a

i

withinYXerror

an

dsquareMean

98.195008.3

53.598

_

_ _ within

meansadjusteds squaremean

squaremeanF

Sokal and Rohlf, 2000. Biometry

Df Sum Sq Mean Sq F value Pr(>F) X 1 4197.0 4197.0 1395.25 < 2.2e-16 ***Group 3 1768.6 589.5 195.98 8.005e-13 ***Residuals 16 48.1 3.0

The output of R: