Week 4 Section: Difference of means, ANOVA, and Matrix Algebra

20 Pages·2014·0.22 MB·English

PS405–Week4Section:
Differenceofmeans,ANOVA,andMatrixAlgebra
D.J.Flynn
February4,2014
t-tests
(cid:73) forequalityoftwosamplemeans
(cid:73) hypotheses:
H :nodifferenceinsamplemeans
0
H :significantdifference
A
(cid:73) calculatingthet-stat:
statistic-hypothesizeddifference
t =
SEofestimate
Gender/partisanshipexample
Question:aremenandwomenequallylikelytobeDemocrats?
t-statfordifferenceinproportions:
(P − P )
m f
t =
(cid:113)
P (1−P ) P (1−P )
m m + f F
n n
m F
p-valuethatRestimatesisfornullofnodifference;confidence
intervalisfordifferencebetweentwosamplemeans
Interpretationofp-value:“ifnullhypothesisistrue,howoften
wouldweobserveadifferencethislargeunderrepeatedsampling?”
–NOTthereisap%chancethatthetruedifferenceisequaltoX.
LogicofANOVAandtheFtest
(cid:73) runningtheme:experimentswith>2groups
(cid:73) doesassignmenttoaparticulargroup(X)affectsome
continuousoutcome(Y)?
(cid:73) thisquestioncanbeansweredwithone-wayANOVA(AKA
F-test)
(cid:73) twosourcesofvariationinDV:
(cid:73) intended:independentvariable/factor
(cid:73) unintended:error/residual
(cid:73) goalofANOVA:determineshareofvarianceexplainedbyX
ANOVAtable
(cid:73) Gothroughtablequickly
(cid:73) Fstatistic(sometimescalledF-act):
explainedvariance MS
A
F = =
unexplainedvariance MS
E
(cid:73) lookupcriticalF-statbasedonnumeratordf,denominatordf,
andconfidencelevel
(cid:73) ifF-act>F-critical,thenwerejectthenullofindependence
ANOVAinR
1. identifyindependentanddependentvariables
2. determinevariablestructures(andchangeifnecessary)
3. estimateANOVAandcallupresults
Determiningvariablestructure
(cid:73) str(variable)returnsthestructureofavariable:integer,
factor,character,number,logical
(cid:73) importantbecauseANOVAsareusedforcategoricalIVs
(cid:73) practice:
install.packages("datasets")
library(datasets)
names(chickwts)
str(chickwts$weight)
str(chickwts$feed)
levels(chickwts$feed)
EstimatingANOVAsinR
anova<-aov(weight∼feed,data=chickwts)
summary(anova)
Df Sum Sq Mean Sq F value Pr(>F)
feed 5 231129 46226 15.37 5.94e-10 ***
Residuals 65 195556 3009
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
Whathappensifweinsteadestimateaov(feed∼weight)?
wrong.model<-aov(feed∼weight,data=chickwts)
Warning messages:
1: In model.response(mf, "numeric") :
using type = "numeric" with a factor response will be
ignored
2: In Ops.factor(y, z$residuals) : - not meaningful
for factors
Anotherexample
Wehavedataonwhichundergraduateinstitutionpeopleattended
andmid-lifesatisfaction(0-100):
names(my.data)
[1] "school" "satisfaction"
table(my.data$school)
school
fsu uf um
5 5 5
my.anova<-aov(satisfaction∼school,data=my.data)
summary(my.anova)
Df Sum Sq Mean Sq F value Pr(>F)
school 2 7216 3608 11.85 0.00144 **
Residuals 12 3655 305
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1