goodness of fit test and test for independence...Øchi-square!distribution Øgoodness!of!fit...

Ø Chi-Square!Distribution

ØGoodness!of!Fit

Ø Examining!Standardized!Residuals

ØTest!for!Independence

ØDifference!of!Two!Proportions

Goodness of Fit Test and Test for Independence

Lecture!15

Section!14.1�14.3,!14.5�14.6

Chi-Square Distribution

• Chi-Square Distribution: continuous!probability!distribution!with!the!following!properties:• Unimodal,!right-skewed,!and!always!non-negative

• Values!get!larger!as!degrees!of!freedom!increase

• Denoted!by!!"# where!$ is!the!number!of!degrees!of!freedom• Degrees!of!freedom!dependent!upon!number!of!categories!in!variable(s)

Note: Because there are many

different shapes, software needs to be

used to calculate probabilities in the

tails, much like the t-distribution.

Chi-Square Table

• Because!the!chi-square!distribution!is!not!symmetric,!the!table!only!gives!statistics!for!areas!in!the!upper!tail.• As!degrees!of!freedom!increase,!chi-square!values!increase!to!get!the!same!area!in!the!upper!tail

• To!reject!the!null!hypothesis,!larger!test!statistics!are!needed!for!tests!that!have!more!degrees!of!freedom.

• Note:!All!chi-square!tests!in!this!course!are!upper!one-sided.

Table!Continues

Motivation: Goodness of Fit Test

• Scenario: CEO!is!interested!in!the!days!of!the!week!that!his!employees!are!most!likely!to!take!off.!!Take!a!random!sample!of!500!employees!and!look!at!the!day!of!the!week!they!last!took!off.

• Question: Do!workers!call!off!work!equally!across!all!five!days?

• Problem: Variable!has!more!than!______________!so!___________________!____________!does!not!apply

• Solution: Compare!________________!against!___________________________!__________________________________________________________________

Day Monday Tuesday Wednesday Thursday Friday

Times Called Off 107 82 68 90 153

Goodness of Fit Test: Hypotheses and Expected Counts

•Hypotheses: Must!be!customized!to!fit!the!problem,!but!the!general!idea!is:• %&:!The!specified!probability!distribution!fits!the!data!well• %':!The!specified!probability!distribution!does!not!fit!the!data!well

• Expected Counts: the!number!of!observations!from!the!sample!that!would!be!expected!to!fall!into!each!category!if!the!distribution!specified!in!the!null!hypothesis!is!true• Hypothesized Proportion: ()• Total Sample Size: *• Expected Count: *()

Example: Goodness of Fit


•Hypotheses:• _______________________________________________________________________________

• _______________________________________________________________________________!______________________



Expected Count 100 100 100 100 100

Goodness of Fit: Conditions

• Conditions:• Count Data: Data!must!be!counts!for!categories!of!a!categorical!variable

• Independence: Counts!in!the!cells!should!be!independent

• Randomization: Individuals!in!table!should!comprise!a!random!sample!from!the!population

• Expected Cell Frequency Count: Expected!number!of!observations!in!each!cell!at!least!5;!that!is,!*() + 5 for!each!assigned!proportion



• Setup:• Count Data: Each!observation!falls!into!__________________________

• Independence: One!employee�s!off-day!likely!_____________________________!__________________________________________________________

• Randomization: Employees!come!from!_________________!and!were!likely!__________________________

• Expected Cell Frequency Count: All!expected!counts!are!________________



Expected Count 100 100 100 100 100

Goodness of Fit: Test Statistic

• To!test!the!hypotheses!in!a!goodness!of!fit!test,!the!test!statistic!is:

,-./# =0)1/

- Observed 2 Expected #

Expectedwhere!3 is!the!number!of!categories!in!the!variable

• Idea: Compare!observed!and!expected!counts!in!each!category• The!_______!the!difference!between!the!observed!and!expected!counts,!the!larger!the!________________

• Larger!test!statistic!means!observed!and!expected!counts!are!_____________

• Reject!null!hypothesis!for!________!test!statistics!à __________________________

Note: Degrees of freedom based on the number of categories; not sample size!



•Mechanics:• Test Statistic:

,# = ___________________________________________________________________________= _____________________________________= __________

• Degrees of Freedom: _____________________

• P-Value: ________________!(Using!software)



Expected Count 100 100 100 100 100

________

________



• Conclusion: With!a!p-value!of!_______________,!we!____________________!______________!and!conclude!that!the!proportion!of!times!employees!called!off!________________________________________________________________.!!Thus,!____________________________________________________________________.



Expected Count 100 100 100 100 100

Standardized Residuals

• Standardized Residual: a!measure!of!how!many!standard!deviations!an!observed!count!is!from!its!corresponding!expected!count!for!a!particular!category

4 = Observed 2 ExpectedExpected

• If!the!null!hypothesis!is!rejected,!standardized!residuals!give!us!an!idea!of!the!categories!that!likely!differed!from!their!hypothesized!proportions.

Example: Standardized Residuals

• Question: What!are!the!standardized!residuals!for!each!day?

• Answer:

Day Observed Expected Standardized Residual

Monday 107 100

Tuesday 82 100

Wednesday 68 100

Thursday 90 100

Friday 153 100


• Question: What!do!the!standardized!residuals!tell!us!about!when!employees!take!off?

• Answer: Compared!to!what!we!would!expect!if!all!days!were!taken!off!equally:!• Monday:!__________________:!_____________________!days!off!than!expected

• Tuesday:!_______________________:!__________days!off!than!expected

• Wednesday:!_______________________:!_________________days!off!than!expected

• Thursday:!__________________:!_____________________!days!off!than!expected

• Friday:!______________________________:!______________!days!off!than!expected


Standardized Res. .70 -1.80 -3.20 -1.00 5.30


• Question: What!do!the!standardized!residuals!tell!us!about!how!the!proportion!of!days!taken!off!relate!to!.20?

• Answer: Proportion!of!days!taken!off!on�• Wednesday!and!Friday!_________________________________!from!.20• __________!residuals

• Tuesday!_______________________________________!from!.20• ______________!residual

• Monday!and!Thursday!likely!______________________________!from!.20• __________!residuals


Standardized Res. .70 -1.80 -3.20 -1.00 5.30

Example: Using Excel

• Scenario:Mars!Company!claims!that!plain!M&M�s!have!the!following!distribution!of!colors:

• Question: Does!Mars�!claim!appear!to!be!accurate?

• Setup:• Hypotheses:

• ______________________________________________________________________________

• ______________________________________________________________________________!___________________________________

Color Blue Orange Green Yellow Brown Red

Hypothesized Proportion .24 .20 .16 .14 .13 .13


• Scenario: After!opening!a!large!bag!of!plain!M&M�s,!you!find!the!following!counts:


• Setup:• Count Data Condition: Each!observation!falls!into!________________________

• Independence: Color!of!one!M&M!has!___________!on!the!color!of!another

• Randomization: Bags!were!__________________________________________________

• Expected Cell Frequency Count: All!expected!counts!are!________________• Verification!on!next!slide


Actual Counts 128 92 83 56 61 49


Note: Both CHISQ.TEST and CHISQ.DIST.RT will calculate the p-value.

They take the following arguments:

• CHISQ.DIST.RT([Test statistic], [Degrees of freedom])

• CHISQ.TEST([Range of observed counts], [Range of expected counts])

B



•Mechanics:• Test Statistic: _______________

• Degrees of Freedom: ___________________

• P-Value: __________

• Conclusion: With!a!p-value!of!_________,!we!fail!to!___________________!_______________!and!conclude!that!the!distribution!of!colors!for!plain!M&M�s!_____________________________________________________.!!Therefore,!Mars!Company�s!claim!___________________________________.


• Question: What!do!you!notice!about!the!standardized!residuals?

• Answer: All!are!______________________• Most!of!the!____________!counts!were!close!to!their!____________!counts

• None!are!_____________________________!to!suggest!any!____________!counts

• Question: Why!should!this!have!been!expected?

• Answer: The!p-value!was!_________,!giving!the!indication!that!the!__________________________________________________________.

• Takeaway: Standardized!residuals!are!more!important!to!analyze!__________________________________________________.


Standardized Residual 1.455 -0.186 0.919 -1.192 0.004 -1.533

Review: Independence

• Scenario: All!employees!at!a!company!are!asked!if!they!would!be!interested!in!attending!a!professional!development!training

• Question: Are!being!a!full!time!employee!and!interest!in!professional!development!independent?

• Answer: ______

• Product of Marginals: ________________________________________________________

• Joint Probability: ______________________________________

Interested Not Interested Total

Full Time 240 60 300

Part Time 160 40 200

Total 400 100 500

Independence: Probability vs. Inference

• In!probability,!two!events!were!deemed!to!be!independent!if!6 7 and 8 = 697: × 698:.• In!inference,!because!we!deal!with!samples!that!are!much!smaller!than!the!population,!it!is!not!practical!to!check!for!exact!equality!between!the!marginal!and!joint!probabilities.

• Instead,!compare!the!observed!counts!against!the!expected!counts!under!the!assumption!the!events!are!independent!to!see!if!they!are!close.

Motivation: Test for Independence

• Scenario: Randomly!sample!400!people!and!record!their!gender!and!handedness

• Question: Are!gender!and!handedness!independent!or!is!one!gender!more!likely!to!be!left!or!right!handed?

• Strategy:• Assume!gender!and!handedness!are!___________________

• Compare!number!of!______________________________________________!against!the!number!we!would!expect!in!the!sample!_____________________________________

Left-Handed Right-Handed Total

Male 20 140 160

Female 20 220 240

Total 40 360 400

Test for Independence: Hypotheses and Expected Counts

•Hypotheses: Inserting!the!variable!names,!the!general!idea!is:• %&:!The!variables!are!independent• %'; The!variables!are!not!independent

• Expected Counts: the!number!of!observations!from!the!sample!that!would!be!expected!to!fall!into!each!cell!in!the!table!if!the!variables!are!actually!independent

<)> =9Row ? sum:9Column @ sum:

Total sample size

Example: Test for Independence

• Question: Are!gender!and!handedness!independent?

•Hypotheses:• __________________________________________________________________

• __________________________________________________________________

Left-Handed Right-Handed Total

Male 20 140 160

Female 20 220 240

Total 40 360 400

Test for Independence: Conditions

• Conditions:• Count Data: Data!must!be!counts!for!categories!of!a!categorical!variable

• Independence: Counts!in!the!cells!should!be!independent

• Randomization: Individuals!in!table!should!comprise!a!random!sample!from!the!population

• Expected Cell Frequency Count: Expected!number!of!observations!in!each!cell!at!least!5



• Conditions:• Count Data: Each!observation!falls!into!__________________________________

• Independence: One!person�s!handedness!__________________________________

• Randomization: Subjects!were!randomly!selected!from!__________________!_______________________________________


Expected Left Right Total

Male 160

Female 240

Total 40 360 400

Test for Independence: Test Statistic

• To!test!hypotheses!in!a!test!for!independence,!the!test!statistic!is:

,9A./:9B./:# =0)1/

- Observed 2 Expected #

Expectedwhere!D is!the!number!of!categories!in!the!row!variable!and!F is!the!number!of!categories!in!the!column!variable

• Idea: Same!as!goodness!of!fit!� compare!observed!and!expected!counts!in!each!category• Larger!test!statistic!means!observed!and!expected!counts!are!far!away• Reject!null!hypothesis!for!large!test!statistics!à Upper!one-sided!test



•Mechanics:• Test Statistic:

,# = _______________________________________________________= ________________________________= __________

• Degrees of Freedom: _________________________________

• P-Value: __________!(Using!software)

Obs. Left Right Total

Male 20 140 160

Female 20 220 240

Total 40 360 400

Exp. Left Right Total

Male 16 144 160

Female 24 216 240

Total 40 360 400

______

________



• Conclusion: With!a!p-value!of!________,!we!___________________________!__________________!and!conclude!gender!and!handedness!_____________!_____________________.!!Thus,!knowing!a!person�s!gender!yields!_______!__________________________________________________.

Obs. Left Right Total

Male 20 140 160

Female 20 220 240

Total 40 360 400

Exp. Left Right Total

Male 16 144 160

Female 24 216 240

Total 40 360 400


• Scenario: Gender!and!handedness!were!found!to!be!independent.

• Question: What!are!the!standardized!residuals?

Gender Handedness Observed Expected Standardized Residual

Male Left 20 16

Male Right 140 144

Female Left 20 24

Female Right 220 216


• Scenario: Gender!and!handedness!were!found!to!be!independent.

• Question: What!do!the!standardized!residuals!tell!us?

• Answer: __________________________________________________• Most!of!the!observed!counts!were!______________!their!expected!counts

• None!are!_________________________________!to!suggest!any!_____________!counts

Std. Res. Left Right

Male 1.00 -0.333

Female -0.816 0.272


• Scenario: Doctors!are!interested!in!patients�!opinions!of!pain!relief!from!three!pain!killers!taken!after!surgery

• Question: Is!the!amount!of!pain!relief!independent!of!pain!killer?

•Hypotheses:• ___________________________________________________________________

• ___________________________________________________________________

Observed Ibuprofen Acetaminophen Codeine Total

Significant Relief 85 80 92 257

Slight Relief 70 83 54 207

Total 155 163 146 464


Note: Both CHISQ.TEST and CHISQ.DIST.RT will calculate the p-value.

• CHISQ.DIST.RT([Test statistic], [Degrees of freedom])

• CHISQ.TEST([Range of observed counts], [Range of expected counts])



• Conditions:• Count Data: Each!observation!falls!into!______________________________

• Independence: One!person�s!pain!__________________________________________

• Randomization: Subjects!likely!all!came!from!________________________,!but!comprise!a!________________________________________





Total 155 163 146 464



•Mechanics:• Test Statistic: _______________

• Degrees of Freedom: ______________________________

• P-Value: __________




Total 155 163 146 464



• Conclusion: With!a!p-value!of!_______,!we!____________________________!and!conclude!the!type!of!pain!killer!and!the!amount!of!pain!relief!a!person!feels!post-surgery!are!_____________________________.

• Question: How!can!we!determine!what!the!relationship!is?

• Answer: Calculate!a!____________________________________________________!_____________________________________




Total 155 163 146 464

Difference of Two Proportions

• To!estimate!the!difference!in!the!proportion!of!successes!between!two!categorical!variables:

G(/ 2 G(# ± H G(/ GI/*/ J G(# GI#

*#

where!H is!the!standard!normal!multiplier!for!confidence!intervals,! G(/ and! G(# are!the!sample!proportions!and!*/ and!*# are!the!respective!sample!sizes

• Note:!The!conditions!are!the!same!as!the!test!for!independence

Example: Difference of Two Proportions

• Scenario: Confidence!intervals!for!every!pair!of!pain!killers!displaying!the!difference!in!the!proportions!is!shown!below

• Question: What!recommendations!would!you!make!to!the!doctor?

• Answer: __________!is!_________________________!at!providing!significant!pain!relief!than!_____________________.!!____________!worked!better!than!_______________,!but!the!difference!was!___________________.!______________!worked!better!than!____________________,!but!this!difference!was!_____!_______________________________

First Medication Second Medication Confidence Interval

Ibuprofen Acetaminophen (-0.052, 0.167)

Codeine Ibuprofen (-0.029, 0.193)

Codeine Acetaminophen (0.030, 0.249)

goodness of fit test and test for independence...Øchi-square!distribution Øgoodness!of!fit...

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