Anova Biometry

8/9/2019 Anova Biometry

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One-way ANOVA• Motivating Example• Analysis of Variance• Model & Assumptions

• Data Estimates of the Model• Analysis of Variance• Multiple Comparisons

• Checking Assumptions• One !ay A"OVA #ransformations


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Motivating Example$#reating Anorexia "ervosa


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Analysis of Variance• Analysis of Variance is a !idely used statistical

techni%ue that partitions the total varia ility in ourdata into components of varia ility that are used totest hypotheses'

• (n One !ay A"OVA) !e !ish to test the hypothesis$ H 0 : µ 1 = µ 2 = = µ k

against$ H a : Not all population means are the same


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Model & Assumptions

• #he model for the o served response is given y$

• *e assume that the errors are normally distri uted

!ith constant variance'

• #his implies that the populations eing sampled are

also normally distri uted !ith e%ual variances+

ijiij x ε α µ ++=


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Analysis of Variance

• (n ANOVA ) !e compare the between-groupvariation !ith the within-group variation to assess!hether there is a difference in the population

means'

• #hus y comparing these t!o measures of variance,spread- !ith one another) !e are a le to detect ifthere are true differences among the underlyinggroup population means'


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• (f the variation et!een the sample means is large)relative to the variation !ithin the samples) then !e!ould e likely to detect significant differences

among the sample means'


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Between Group Variation is LargeCompared to Within Group Variation

Here we would almost certainly reject the null hypothesis.


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Analysis of Variance Between-group

variation is largecompared to theWithin-group

variation

µ 2

µ 3

µ 1

µ α 3

α 2 = µ 2 - µ

α 1

If we sampled from

these populations, wewould expect to rejectH0

ε 3 j = y3 j − µ 3


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• (f the variation et!een the sample means is small)relative to the variation !ithin the samples) thenthere !ould e considera le overlap of

o servations in the different samples) and !e !oulde unlikely to detect any differences among thepopulation means'


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Between Group Variation is SmallCompared to Within Group Variation

Here we would fail to reject the null hypothesis.


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µ . µ / . µ 0 . µ 1

All α i = 0If we sampled

from thesepopulations, we

would not expectto reject H 0

ε 2 j = y2 j - µ 2


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Analysis of Variance• (f !e consider all of the data together) regardless of

!hich sample the o servation elongs to) !e canmeasure the overall total varia ility in the data y$

• #his is the Total Sum of S uares , SS Total -'

• (f !e divide this sum of s%uares y its degrees offreedom , N − /-) !e !ill have a measure ofvariance'

∑∑= = ••−k

i

n

jij

i

x x1 1

2

)(


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Analysis of Variance• "o!) the deviation of every o servation from the overall

,grand- mean can e partitioned as$

• 2%uaring and summing across all o servations)

!e get$

iijiij x x x x x x −+−=− •••••• )()()(

∑∑ ∑∑∑∑= = = =

•= =

••••• −+−=−k

i

n

j

k

i

n

jiij

k

i

n

jiij

i ii

x x x x x x1 1 1 1

2

1 1

22 )()()(

Measure variation dueto the fact differenttreatments are used.

Measures errorvariation, variation inresponse when same

treatment is applied.


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Analysis of Variance• "o!) the deviation of every o servation from the overall

,grand- mean can e partitioned as$

• 2%uaring and summing across all o servations)

!e get$

iijiij x x x x x x −+−=− •••••• )()()(

∑∑ ∑∑∑∑= = = =

•= =

••••• −+−=−k

i

n

j

k

i

n

jiij

k

i

n

jiij

i ii

x x x x x x1 1 1 1

2

1 1

22 )()()(

Treatment Sum of Squares(SS Treat ) or Between GroupSum of Squares

Error Sum of Squares (SS Error ) orWithin Group Sum of Squares


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• #o convert Sums of S uares , SS - into compara lemeasures of variance) !e need to divide the SS ytheir respective degrees of freedom '

• #his gives us mean s uares , MS - !hich aremeasures of variance $

MS Treat = SS Treat / df Treat = SS Treat / (k – 1) MS rror = SS rror / df rror = SS rror / ( ! – k )


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Analysis of Variance• #he expected values of the mean s%uares for

repeated sampling are$ E( MS Treat ) = σ 2 + Σα i2 / (k − 1)

E( MS rror ) = σ 2

• #hus MS Error is an estimate of σ 0 ) the

!ithin group variance$

• (f all the α i are 3) then the expected value for the!-ratio !ill e σ 0 4 σ 0 . /) !hile if some of the α i are

not 3) E, MS B - 5 E, MS W -) and E, F - 5 /

ˆW MS = 2σ and σ ̂=W MS


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Analysis of Variance "# • Our test statistic is the !-ratio ,or !-statistic -

!hich compares these t!o mean s%uares$

"ote that the greater the natural varia ility !ithinthe groups) the larger the effects , α i - !ill need toe ,as estimated y MS

Treat - for us to detect any

significant differences'

rror

Treat

MS MS

" =0


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Analysis of Variance• #raditionally the Analysis of Variance calculations

have een presented in an ANOVA Table '• #he format of the ta le is$

k – 1

Source of e!rees of Sum of Mean F- "atio P-value#ariation $reedom Squares SquareTreatment SS Treat MS Treat MS Treat /MS rror Tail AreaError ! – k SS

rror MS

rror Total ! – 1 SS T

hese cols add up !!" df


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Motivating $%ample

# 0 = $0%&$'"(%&)* = (&+'


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Analysis of Variance• A large !-statistic provides evidence against H 3

!hile a small !-statistic indicates that the data andH 3 are compati le'

• #o calculate a -value to test H 3 , !e compare the!-statistic !e o tained from our data to thedistri ution it !ould have under a true H 3 ) i'e' an!-distribution !ith ,k − /- and ," – k - degrees of

freedom '

• "ote that F 3 is al!ays positive) so this is al!ays aone tailed test'


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hen the -value = 0&0)

When H 0 is true, # 0 # .df / ,df '

1et2s sa3 our o4served value for # was # 0 = '&(

3 / 0 1 6

3 ' 3

3 ' 0

3 ' 6

3 ' 7

3 ' 8

F-distribution

#or example, consider the # -distri4ution with + and $0 df


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Multiple 'omparisons• A significant !-test tells us that at least t!o of the

underlying population means are different) ut itdoes not tell us !hich ones differ from the others'

• *e need extra tests to compare all the means)

!hich !e call Multiple Comparisons'• *e look at the difference et!een every pair of

group population means) as !ell as the confidenceinterval for each difference'

• *hen !e have ( groups) there are$

possi le pair !ise comparisons'5 choose ' ( )

21

)2(22

−=−=

k k

k k k


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Multiple 'omparisons • (f !e estimate each comparison separately !ith 9:;

confidence) the overall error rate !ill e greaterthan )* '

• 2o) using ordinary pair !ise comparisons ,i'e' lotsof individual t-tests -) !e tend to find too manysignificant differences et!een our sample means'

• *e need to modify our intervals so that theysimultaneously contain the true differences !ith 9:;

confidence across the entire set of comparisons'• #he modified intervals are kno!n as$

simultaneous confidence intervals O<multiple comparison procedures


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Multiple 'omparisons• =irst) the +onferroni correction '• (nstead of using t df, α / 2 as our multiplier for the

confidence interval) !e use t df ,α / 2L ) !here is thetotal num er of possi le pair !ise comparisons

,i'e' = ( ( !" / 2 -'• #hat is) !e divide α 40 y the num er of tests to e

done , α 40L-'• #his assumes all pair !ise comparisons are

independent) !hich is not the case) so thisad>ustment is too conservative ,intervals !ill e too!ide? i'e' finds too fe! significant differences-'


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Multiple 'omparisons• 2econd) !e have Tu(ey #ntervals '• #he calculation of #ukey (ntervals is %uite

complicated) ut overcomes the pro lems of theunad>usted pair !ise comparisons finding too many

significant differences,i'e' confidence intervals that are too narro!-)and the @onferroni correction finding too fe!significant differences,i'e' confidence intervals that are too !ide-'

We will use u5e3 Intervals


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#ukey air !ise Comparisons

Select Compare Means > All Pairs, Tukey HSD

!ere "e see that onl# $eha%ioral an& 'tan&ar& therapies &i erin terms o mean "ei ht ain* e estimate those in ,eha%ioraltherap# "ill ain ,et"een 2 l,s* an& 13 l,s* more on a%era e*


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'hec(ing Assumptions. #ndependence

• #he o servations !ithin each sample must eindependent of one another'

• #he samples must e taken from independentpopulations'

'h (i A i


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'hec(ing Assumptions.$ uality of Variance #

• E%uality of variance is very important in One !ay A"OVA'

• *e check e%uality of variance using Bevene s)@artlett s) @ro!n =orsythe) or O @rien s #ests '

• (f the assumption of e%ual population variances isnot satisfied , small -value from these tests-) !ecan try transforming the data or use *elch s

A"OVA !hich allo!s the variances to e une%ual'

'hec(ing Assumptions


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'hec(ing Assumptions.$ uality of Variance

• =or many data sets) !e often find there is arelationship et!een the centre of the dataand the spread of the data$• (n particular) samples !ith lo! means ,or

medians- often have small spread !hile samples!ith large means ,or medians- often have largespread ,or vice versa-'

• #he positive relationship et!een the mean andvariance ,or et!een the median and midspread-in different samples is often true for data thathave right ske!ed distri utions'

'hec(ing Assumptions


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hec(ing Assumptions.$ uality of Variance

• (f the variance of the samples isincreasing as the sample means

increase a log or s%uare roottransformation is often times used'


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One-way ANOVA Transformations• *e can transform our response varia le if !e

detect pro lems !ith the e%uality of variance ornormality assumptions'

• o!ever) as in the t!o sample situation) !e canonly use a log transformation if !e !ish to e a leto ack transform and interpret our confidenceintervals meaningfully'

Anova Biometry

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