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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
• Question: Do!workers!call!off!work!equally!across!all!five!days?
•Hypotheses:• _______________________________________________________________________________
• _______________________________________________________________________________!______________________
Day Monday Tuesday Wednesday Thursday Friday
Times Called Off 107 82 68 90 153
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
Example: Goodness of Fit
• Question: Do!workers!call!off!work!equally!across!all!five!days?
• 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!________________
Day Monday Tuesday Wednesday Thursday Friday
Times Called Off 107 82 68 90 153
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!
Example: Goodness of Fit
• Question: Do!workers!call!off!work!equally!across!all!five!days?
•Mechanics:• Test Statistic:
,# = ___________________________________________________________________________= _____________________________________= __________
• Degrees of Freedom: _____________________
• P-Value: ________________!(Using!software)
Day Monday Tuesday Wednesday Thursday Friday
Times Called Off 107 82 68 90 153
Expected Count 100 100 100 100 100
________
________
Example: Goodness of Fit
• Question: Do!workers!call!off!work!equally!across!all!five!days?
• Conclusion: With!a!p-value!of!_______________,!we!____________________!______________!and!conclude!that!the!proportion!of!times!employees!called!off!________________________________________________________________.!!Thus,!____________________________________________________________________.
Day Monday Tuesday Wednesday Thursday Friday
Times Called Off 107 82 68 90 153
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
Example: Standardized Residuals
• 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
Day Monday Tuesday Wednesday Thursday Friday
Standardized Res. .70 -1.80 -3.20 -1.00 5.30
Example: Standardized Residuals
• 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
Day Monday Tuesday Wednesday Thursday Friday
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
Example: Using Excel
• Scenario: After!opening!a!large!bag!of!plain!M&M�s,!you!find!the!following!counts:
• Question: Does!Mars�!claim!appear!to!be!accurate?
• 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
Color Blue Orange Green Yellow Brown Red
Actual Counts 128 92 83 56 61 49
Example: Using Excel
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
Example: Using Excel
• Question: Does!Mars�!claim!appear!to!be!accurate?
•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!___________________________________.
Example: Standardized Residuals
• 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!__________________________________________________.
Color Blue Orange Green Yellow Brown Red
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
Example: Test for Independence
• Question: Are!gender!and!handedness!independent?
• Conditions:• Count Data: Each!observation!falls!into!__________________________________
• Independence: One!person�s!handedness!__________________________________
• Randomization: Subjects!were!randomly!selected!from!__________________!_______________________________________
• Expected Cell Frequency Count: All!expected!counts!are!________________
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
Example: Test for Independence
• Question: Are!gender!and!handedness!independent?
•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
______
________
Example: Test for Independence
• Question: Are!gender!and!handedness!independent?
• 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
Example: Standardized Residuals
• 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
Example: Standardized Residuals
• 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
Example: Test for Independence
• 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
Example: Using Excel
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])
Example: Test for Independence
• Question: Is!the!amount!of!pain!relief!independent!of!pain!killer?
• Conditions:• Count Data: Each!observation!falls!into!______________________________
• Independence: One!person�s!pain!__________________________________________
• Randomization: Subjects!likely!all!came!from!________________________,!but!comprise!a!________________________________________
• Expected Cell Frequency Count: All!expected!counts!are!________________
Observed Ibuprofen Acetaminophen Codeine Total
Significant Relief 85 80 92 257
Slight Relief 70 83 54 207
Total 155 163 146 464
Example: Test for Independence
• Question: Is!the!amount!of!pain!relief!independent!of!pain!killer?
•Mechanics:• Test Statistic: _______________
• Degrees of Freedom: ______________________________
• P-Value: __________
Observed Ibuprofen Acetaminophen Codeine Total
Significant Relief 85 80 92 257
Slight Relief 70 83 54 207
Total 155 163 146 464
Example: Test for Independence
• Question: Is!the!amount!of!pain!relief!independent!of!pain!killer?
• 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!____________________________________________________!_____________________________________
Observed Ibuprofen Acetaminophen Codeine Total
Significant Relief 85 80 92 257
Slight Relief 70 83 54 207
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)