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ANOVA 1 fator - Modelo com efeito aleatrio - Exemplo

Richard A. Armstrong; F. Eperjesi and B. Gilmartin - Application of ANOVA in Optometry

Opthtal. Physiol. Opt. 2002 22: 248-256.

Experimento: delineamento inteiramente casualizado

O objetivo estimar o grau de variao de uma medida particular e em muitos casos comparar diferentes fontes de variao em espao e em tempo.

Neste caso, neste exemplo, as componentes de varincia so mais teis que o teste F para indicar que a variao entre dias quase duas vezes aquela num nico dia.

One-way ANOVA (random effects model). An alternative one-way model is the random effects

model in which the objective is not to measure a fixed effect but to estimate the degree of variation of a

particular measurement and to compare different sources of variation in space and/or time. These designs

are often called nested or hierarchical designs. Hence, in our glaucoma trial, we may wish to determine

the degree of variability in the response to a treatment within a single patient, between different patients,

or with time. The most important statistics from a random effects model are the components of variance

which estimate the variance associated with each of these sources of variation influencing a

measurement. Hence, the nested design is particularly useful in preliminary experiments

designed to estimate different sources of variation and hence, in the design of experiments.

Statistical guidelines for clinical studies of human vision. Richard A Armstrong, Leon N Davies, Mark C M Dunne and Bernard Gilmartin. Ophthalmic & Physiological Optics 31 (2011) 123136.

n = 1 paciente medida = IOP = presso intra-ocular

trs sesses de medies distanciadas de 3 dias cada sesso Days = dias 3 ; 6 e 9 dias. Foram efetuadas 5 medies no mesmo indivduo:

- aps cada 1 min foi efetuada sucessivamente uma leitura da presso e, tambm, no 3 dia; no 6 dia e no 9 dia.

Valores

Sesso 1 Sesso 2 Sesso 3 18 17 19 19 18 18 20 16 20 19 17 20 21 17 19

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disposio dos valores numa nica coluna C1 C2

IOP = presso intra-ocular Days 18 1 19 1 20 1 19 1 21 1 17 2 18 2 16 2 17 2 17 2 19 3 18 3 20 3 20 3 19 3

Comandos no Minitab

Aps Ctrl+L digitamos no espao em branco ANOVA 'IOP' = Days; Random 'Days';

EMS; Means Days.

Ou

ANOVA c1 = c2; Random c2;

EMS; Means c2.

Ou via Menu do Minitab Stat >> ANOVA >> Balanced ANOVA

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Ou via Menu do Minitab Stat >> ANOVA >> Balanced ANOVA

Clicar em Results e check box (X) Display expected mean squares and variance components

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Clicamos em OK

ANOVA: IOP versus Days Factor Type Levels Values Days random 3 1, 2, 3 Analysis of Variance for IOP Source DF SS MS F P Days 2 17.7333 8.8667 10.64 0.002 Error 12 10.0000 0.8333 Total 14 27.7333 S = 0.912871 R-Sq = 63.94% R-Sq(adj) = 57.93%

No STATISTICA for Windows -- ANOVA one-way com o fator Days como Var. Independente

Univariate Tests of Significance for IOP (Spreadsheet1) Sigma-restricted parameterization Effective hypothesis decomposition

SS Degr. of - Freedom MS F p Intercept 5152.267 1 5152.267 6182.720 0.000000 Days 17.733 2 8.867 10.640 0.002198 Error 10.000 12 0.833

among-group variance = (MSamongMSwithin)/n

Componente da Varincia para Days = (8.866 -0.833) / 5 = 1.607

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Expected Mean Square for Each Term (using Variance Error unrestricted Source component term model) 1 Days 1.6067 2 (2) + 5 (1) 2 Error 0.8333 (2)

Error = variao de 1 min a 1 min

Relativa Contribuio da Varincia para Dias

= (1.6067 /(1.60678+0.8333) = (1.6067 / 2.44)*100 = 65.85% Means Days N IOP 1 5 19.400 2 5 17.000 3 5 19.200

.......................................................................................................................................

Concluso:

Expected Mean Square for Each Term (using Variance Error unrestricted Source component term model) 1 Days 1.6067 2 (2) + 5 (1) 2 Error 0.8333 (2)

prefervel medir uma vez a cada 3 dias do que medir num nico dia vrias vezes !

aconselhvel medir a presso s uma vez, mas numa amostra de dias e no medir seguidamente a cada 1 minuto

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A variao a cada 3 dias = 1.6067 o dobro da variao a cada 1 minuto num dia = 0.833

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Se IOP variasse consideravelmente de minuto a minuto, mas em mdia pouco entre os dias, ento, a melhor estratgia seria obter vrias medies

de IOP num nico dia.

Se a variao de minuto a minuto fosse desprezvel, mas houvesse variao significante de dia para dia, seria melhor medir IOP s uma vez, mas numa

amostra de dias. o caso desse nosso exemplo: - pois a variao a cada 3 dias = 1.6067 o

dobro da variao a cada 1 minuto num dia = 0.833.

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http://udel.edu/~mcdonald/statanovapart.html

Handbook of Biological Statistics

Componente de varincia para Planejar o Experimento

Another area where partitioning variance components is useful is in

designing experiments.

For example, let's say you're planning a big experiment to test the effect of

different drugs on calcium uptake in rat kidney cells. You want to know how many

rats to use, and how many measurements to make on each rat, so you do a pilot

experiment in which you measure calcium uptake on 6 rats, with 4 measurements per

rat. You analyze the data with a one-way anova and look at the variance

components.

If a high percentage of the variation is among rats that would tell you that

there's a lot of variation from one rat to the next, but the measurements within one

rat are pretty uniform. You could then design your big experiment to include a lot of

rats for each drug treatment, but not very many measurements on each rat. Or you

could do some more pilot experiments to try to figure out why there's so much rat-to-rat

variation (maybe the rats are different ages, or some have eaten more recently than

others, or some have exercised more) and try to control it.

On the other hand, if the among-rat portion of the variance was low, that would tell you that the mean values for different rats were all about the same, while there

was a lot of variation among the measurements on each rat. You could design your big experiment with fewer rats and more observations per rat, or you could try to

figure out why there's so much variation among measurements and control it better. .................................