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Analysis of Covariance. Goals:. Reduce error variance. Remove sources of bias from experiment. Obtain adjusted estimates of population means. Analysis of Covariance. Y. Y. X .1. X . X .2. X. X .1. X. X .2. X . Random assignment conditional distribution is less variable.

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Presentation Transcript
slide1

Analysis of Covariance

Goals:

  • Reduce error variance.
  • Remove sources of bias from experiment.
  • Obtain adjusted estimates of population means.
slide2

Analysis of Covariance

Y

Y

X.1

X..

X.2

X

X.1

X

X.2

X..

Random assignment

conditional distribution

is less variable

Not randomly assigned

Need to block

slide3

Analysis of Covariance

Selection of covariates:

  • Experiment contains one or more nuisance, uncontrollable, variables that we know.
  • Control of these variables is either not possible or not feasible.
  • It is possible to obtain a measure of the nuisance variable that doesn’t include effects attributable to the factors: a) the covariate is obtained prior to presentation of factor levels, b) the covariates are obtained after the levels are set but before they affect, c) it can be assumed that they are not affected by the factor levels. (These are not always necessary).
slide6

Analysis of Covariance

Some more assumptions for the above model:

  • The slope b does not equal zero.
  • The relationship between Yij and Xij is linear.
  • The regression coefficients for each factor level are equal.
  • The covariate variable Xij is not affected by the levels of the factor.
slide8

Analysis of Covariance

Back to 401

  • Simplest, most reduced model:

The fitted model:

slide9

Analysis of Covariance

  • Simplest, most reduced model (Cont.):

Residual:

and

Difference from Grand Mean.

Total error we still need to explain.

Degrees of Freedom for this error:

slide10

Analysis of Covariance

How to perform the analysis of covariance:

  • Wanted to explain more, added the factor. The linear model became:

The fitted model:

slide11

Analysis of Covariance

  • Cont.

Residual:

How much did we explain from the original total residual?

Original total residual.

Part of residual explained by Factor A.

Part of residual still not accounted for.

=

Syy

-

Eyy

Tyy

slide12

Analysis of Covariance

  • Cont.

Rearrange:

Total original Residual

Part of residual explained by Factor A.

Part of residual still not accounted for

Syy

Tyy

Eyy

New: ANCOVA

Old: ANOVA

SST

SSA

SSE

df

slide13

Analysis of Covariance

  • Adding the covariate:

The fitted model:

slide14

Analysis of Covariance

  • Cont.

Can show that:

slide17

Analysis of Covariance

  • Cont.

Residual:

How much did we explain from the original total residual?

Part of residual explained by covariate

Part of residual explained by factor A

Part of residual not accounted for after adjusting using the covariate and the factor

Total original Residual

slide18

Analysis of Covariance

  • Cont.

Rearrange:

explained by covariate

explained by factor

not accounted for by full model

Total original Residual

df

slide19

Analysis of Covariance

Lets go the other route:

  • Simplest, most reduced model:

The fitted model:

slide20

Analysis of Covariance

  • Simplest, most reduced model (Cont.):

Residual:

and

Difference from Grand Mean.

Total error we still need to explain.

Degrees of Freedom for this error:

slide21

Analysis of Covariance

  • Want to explain more, add covariate. The linear model became:

The fitted model:

slide22

Analysis of Covariance

  • Cont.

Can show that:

slide24

Analysis of Covariance

  • Cont.

Residual:

How much did we explain from the original total residual?

Original total residual.

Part of residual still not accounted for.

Part of residual explained by covariate.

slide25

Analysis of Covariance

  • Cont.

Rearrange:

Total original Residual

Explained by covariate.

not accounted for by regression

df

slide26

Analysis of Covariance

  • Compare the errors:

Part of residual not accounted for after adjusting using the covariate and the factor

Along with its df.

Part of residual not accounted for after regression along with its df.

slide27

Analysis of Covariance

  • Compare the errors (Cont.):

The difference is what the factor accounts for.

slide28

Analysis of Covariance

A study to compare the strength of monofilament fiber produced by three different machines:

Three factors: Machine 1 (m1), Machine 2 (m2), and Machine 3 (m3).

Covariate: is the thickness of the fiber.