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ロジスティック回帰 Part 2 - 係数、オッズ比、平均限界効果
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Kan Nishida
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September 26, 2019
Science
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ロジスティック回帰 Part 2 - 係数、オッズ比、平均限界効果
Kan Nishida
PRO
September 26, 2019
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Transcript
ϩδεςΟοΫճؼ Part 2 ɺΦοζൺɺฏۉݶքޮՌ Exploratory Seminar #20
EXPLORATORY
3 εϐʔΧʔ ా צҰ CEO EXPLORATORY ུྺ 2016ɺσʔλαΠΤϯεͷຽओԽͷͨΊɺExploratory, Inc Λ
্ཱͪ͛Δɻ Exploratory, Inc.ͰCEOΛΊΔ͔ͨΘΒɺσʔλαΠΤϯεɾ ϒʔτΩϟϯϓɾτϨʔχϯάͳͲΛ௨ͯ͠γϦίϯόϨʔͰ ߦΘΕ͍ͯΔ࠷ઌͷσʔλαΠΤϯεͷීٴͱڭҭʹऔΓ Ήɻ ถΦϥΫϧຊࣾͰɺ16ʹΘͨΓσʔλαΠΤϯεͷ։ൃνʔ ϜΛ͍ɺػցֶशɺϏοάɾσʔλɺϏδωεɾΠϯςϦδΣ ϯεɺσʔλϕʔεʹؔ͢Δଟ͘ͷΛੈʹૹΓग़ͨ͠ɻ @KanAugust
Vision ΑΓΑ͍ҙࢥܾఆΛ͢ΔͨΊʹ σʔλΛ͏͜ͱ͕ͨΓલʹͳΔ
Mission σʔλαΠΤϯεͷຽओԽ
6 ୈ̏ͷ σʔλαΠΤϯεɺAIɺػցֶश౷ܭֶऀɺ։ൃऀͷͨΊ͚ͩͷͷͰ͋Γ·ͤΜɻ σʔλʹڵຯͷ͋ΔਓͳΒ୭͕ੈքͰ࠷ઌͷΞϧΰϦζϜΛͬͯ ϏδωεσʔλΛ؆୯ʹੳͰ͖Δ͖Ͱ͢ɻ Exploratory͕ͦ͏ͨ͠ੈքΛՄೳʹ͠·͢ɻ
ୈ1ͷ ୈ̎ͷ ୈ̏ͷ ϓϥΠϕʔτ(ߴ͍/ݹ͍) Φʔϓϯɾιʔε(ແྉ/࠷ઌ) UI & ϓϩάϥϛϯά ϓϩάϥϛϯά 2016
2000 1976 ϚωλΠθʔγϣϯ ίϞσΟςΟԽ ຽओԽ ౷ܭֶऀ σʔλαΠΤϯςΟετ Exploratory ΞϧΰϦζϜ Ϣʔβʔɾ ମݧ πʔϧ Φʔϓϯɾιʔε(ແྉ/࠷ઌ) UI & ࣗಈԽ ϏδωεɾϢʔβʔ ςʔϚ σʔλαΠΤϯεͷຽओԽ
質問 ExploratoryɹϞμϯˍγϯϓϧ UI 伝える データアクセス データ ラングリング 可視化 アナリティクス 統計/機械学習
ϩδεςΟοΫճؼ Part 2 ɺΦοζൺɺฏۉݶքޮՌ Exploratory Seminar #20
質問 伝える データアクセス データ ラングリング 可視化 アナリティクス 統計/機械学習
USͷͪΌΜσʔλ
ڵຯͷର ΧςΰϦʔ/ೋ߲ 12 ΧςΰϦʔ/ଟ߲
• ͷྸ͍͔ͭ͘ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ • 35ࡀΑΓ্ͳͷ͔ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ
• ͷྸ͍͔ͭ͘ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ • 35ࡀΑΓ্ͳͷ͔ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ
ڵຯͷର ΧςΰϦʔ/ೋ߲ 15 ΧςΰϦʔ/ଟ߲
ઢܗճؼ
17 Father_Age = a * Mother_Age + b ʢ͖ʣ ย
ઢܗճؼͷϞσϧʢܭࢉࣜʣ
18 Father_Age = a * Mother_Age + b ʢ͖ʣ ย
ͱยΛௐઅ͢Δ͜ͱͰ࣮σʔλͱ Ϛον͢ΔΑ͏ͳઢ͕ඳ͚Δɻ
19 ʢ͖ʣ ย
20 Father_Age = 0.87 * Mother_Age + 6.28 ʢ͖ʣ ย
ઢܗճؼͷϞσϧʢܭࢉࣜʣ
None
ͷྸ ͷྸ ͷྸ͕1্͕Δͱɺͷྸ0.87্͕Δɻ
ͷྸ ͷྸ ઢܗճؼͷϞσϧ࣮σʔλͱϑΟοτ͢ΔΑ͏ʹ࡞ΒΕΔɻ
• ͷྸ͍͔ͭ͘ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ • 35ࡀΑΓ্ͳͷ͔ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ
ڵຯͷର ΧςΰϦʔ/ೋ߲ 25 ΧςΰϦʔ/ଟ߲
• ͜ͷϢʔβʔίϯόʔτ͢Δ͔ʁ • ͜ͷऔҾෆਖ਼͔ʁ • ͜ͷैۀһΊΔ͔ʁ • ͜ͷͪΌΜະख़ࣇͰੜ·ΕΔ͔ʁ ೋ߲ͷ࣭
27 ͕35Ҏ্ͷ֬ = logistic(a * Mother_Age + b) ʢ͖ʣ ย
ϩδεςΟοΫճؼͷϞσϧʢܭࢉࣜʣ
28 ͕35Ҏ্ͷ֬ = logistic(a * Mother_Age + b) ʢ͖ʣ ย
ͱยΛௐઅ͢Δ͜ͱͰ࣮σʔλͱ Ϛον͢ΔΑ͏ͳۂઢ͕ඳ͚Δɻ
࣮σʔλ
දܭࢉͷʮׂ߹ʢˋ of ߹ܭʣʯΛͬͯ TRUE/FALSEͷׂ߹Λදࣔ͢Δɻ
ͷྸ͝ͱͷTRUE/FALSEͷׂ߹
ຌྫͷதͷFALSEΛΫϦοΫͯ͠ɺFALSEͷ෦ͷόʔΛফ͢ɻ
ଞʹʢͬͱ؆୯ʹʣಉ͡Α͏ͳ νϟʔτΛඳ͘ํ๏͕͋Δɻ
Y࣠ʹϩδΧϧܕͷྻΛબͼʮ% of TRUEʯͷܭࢉΛબͿɻ
ϥΠϯνϟʔτʹม͑ͯΈΔɻ
͜ͷ࣮σʔλʹϑΟοτ͢ΔϩδεςΟοΫۂઢΛग़͍ͨ͠ɻ
37 ͕35Ҏ্ͷ֬ = logistic(a * Mother_Age + b) ʢ͖ʣ ย
ͱยΛௐઅ͢Δ͜ͱͰ࣮σʔλͱ Ϛον͢ΔΑ͏ͳۂઢ͕ඳ͚Δɻ
38 ϩδεςΟοΫճؼͷϞσϧ
39 ͕35Ҏ্ͷ֬ = logistic(0.29 * Mother_Age - 10.12) ย
None
ϩδεςΟοΫճؼʹΑΔ༧ଌͷྻΛY࣠ʹׂΓͯɺ ʮฏۉʯͷܭࢉΛબͿɻ
ϩδεςΟοΫճؼʹΑΔ༧ଌ0͔Β1ͷؒͷͳͷͰɺ Y2࣠ʹׂΓͯΔɻ
࣮σʔλ Ϟσϧ (ϩδεςΟοΫۂઢ) ͍͍ײ͡Ͱ࣮σʔλʹϑΟοτͯ͠Δɻ
ͱ͜ΖͰɺ͜ͷۂઢɺͲ͏ղऍͨ͠Β͍͍ͷ͔ʁ P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12)
45 ϩδεςΟοΫճؼ ༧ଌมͷӨڹͷղऍ
46 ϩδεςΟοΫճؼ • ʢCoefficientʣ • ΦοζൺʢOdds Ratioʣ • ฏۉݶքޮՌʢAverage Marginal
Effectʣ
47 ϩδεςΟοΫճؼ • ʢCoefficientʣ • ΦοζൺʢOdds Ratioʣ • ฏۉݶքޮՌʢAverage Marginal
Effectʣ
48 มͷࢦඪͱͯ͠ɺΛબ͢Δɻ
None
None
͕খ͍͞ͱɺ༧ଌม ͇ͷͷมԽ͕͈ͷ֬ ͷมԽʹ͋ͨ͑ΔӨڹ ͕খ͍͞ɻ 51 y = logistic(0.1 * x)
͕େ͖͍ͱɺ༧ଌม ͇ͷͷมԽ͕͈ͷ֬ ͷมԽʹ͋ͨ͑ΔӨڹ ͕େ͖͍ɻ 52 y = logistic(10 * x)
P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12)
P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12) Pr(Father
> 35) = Logit (0.29 * Mother_Age - 10.12) -1
Logit( P(Father > 35) ) = 0.29 * Mother_Age -
10.12 P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12) P(Father > 35) = (0.29 * Mother_Age - 10.12) Logit -1
ϩδοτؔ֬ΛϩάɾΦοζม͢Δ Logit( P(y) ) = Log(Odds(y)) Logit( P(Father > 35)
) = 0.29 * Mother_Age - 10.12 Log(Odds(Father > 35)) = 0.29 * Mother_Age - 10.12
Log(Odds((Father > 35))) = 0.29 * 20 - 10.12 =
-4.32 ͕20 Log(Odds(Father > 35)) = 0.29 * Mother_Age - 10.12
Log(Odds((Father > 35))) = 0.29 * 20 - 10.12 =
-4.32 ͕21 Log(Odds((Father > 35))) = 0.29 * 21 - 10.12 = -4.03 ͕20 Log(Odds(Father > 35)) = 0.29 * Mother_Age - 10.12
Log(Odds((Father > 35))) = 0.29 * 20 - 10.12 =
-4.32 ͕21 Log(Odds((Father > 35))) = 0.29 * 21 - 10.12 = -4.03 ͕20 Log(Odds(Father > 35)) = 0.29 * Mother_Age - 10.12 0.29 ࠩ ͷྸ͕1ࡀ্͕Δͱɺ͕35ࡀҎ্Ͱ͋Δ ϩάɾΦοζ͕0.29্͕Δɻ
ϩάɾΦοζͬͯԿ͚ͩͬʁ
͏গ͠ਓؒతͳࢦඪ͕͋Δɻ
62 ϩδεςΟοΫճؼ • ʢCoefficientʣ • ΦοζൺʢOdds Ratioʣ • ฏۉݶքޮՌʢAverage Marginal
Effectʣ
None
64 Φοζൺ (Coefficient) ʹࢦؔ(logͷٯ)Λద༻ͨ͠ɻ Φοζൺ = exp()
65 ͕35Ҏ্ͷ֬ = logistic(a * Mother_Age + b) ʢ͖ʣ ย
ͱยΛௐઅ͢Δ͜ͱͰ࣮σʔλͱ Ϛον͢ΔΑ͏ͳۂઢ͕ඳ͚Δɻ
66 ϩδεςΟοΫճؼͷϞσϧ
67 ͕35Ҏ্ͷ֬ = logistic(0.29 * Mother_Age - 10.12) ย
None
֬ (Father > 35) ͷྸ
ϩδεςΟοΫۂઢ
ϩδεςΟοΫۂઢ͔ΒΦοζΛܭࢉͯ͠ΈΔɻ
72 Φοζ Φοζ = TRUEͷ֬ / FALSEͷ֬
73 ૣ࢈ʹͳΔΦοζ Φοζ = TRUEͷ֬ / FALSEͷ֬ ૣ࢈ʹͳΔ͕֬10% ૣ࢈ʹͳΒͳ͍͕֬90% 10
/ 90 = 0.1111…
74 50% 50% 100% 0% mother_age(ͷྸ) 34 When Mother is
34, what is the odds of Father being older than 35?
75 Φοζ 1 50% 50% 50/50 100% 0% mother_age(ͷྸ) 34
76 Φοζ 1 50% 50% 50/50 34 mother_age(ͷྸ) 100% 0%
77 1 50% 50% 66.7/33.3 2 33.3% 66.7% 34 35
Φοζ mother_age(ͷྸ) 100% 0%
78 1 50% 50% 80/20 2 33.3% 66.7% 34 35
20% 80% 36 4 Φοζ mother_age(ͷྸ) 100% 0%
79 1 50% 50% 88.9/11.1 33.3% 66.7% 34 35 20%
80% 36 11.1% 88.9% 37 2 4 8 Φοζ mother_age(ͷྸ) 100% 0%
80 มͷ͕1૿͑ΔͱɺΦοζԿഒʹͳΔ͔ɻ Φοζൺ (Odds Ratio)
81 TRUE FALSE 1 50% 50% 33.3% 66.7% 20% 80%
11.1% 88.9% 2 4 8 Φοζ 2x Φοζൺ mother_age(ͷྸ) 34 35 36 37
82 TRUE FALSE 1 50% 50% 33.3% 66.7% 20% 80%
11.1% 88.9% 2 4 8 Φοζ 2x Φοζൺ mother_age(ͷྸ)͕ 1্͕Δͱŋŋŋ TRUEͱͳΔΦοζ͕2ഒʹͳΔɻ mother_age(ͷྸ) 34 35 36 37
83 TRUE FALSE 1 50% 50% 33.3% 66.7% 20% 80%
11.1% 88.9% 2 4 8 Φοζ 2x Φοζൺ mother_age(ͷྸ) 34 35 36 37 Logistic Curve guarantee that this Odds Ratio is constant.
ม͕ΧςΰϦʔͷ࣌Ͳ͏ղऍ͢ΕΑ͍͔ɻ
༧ଌม͕ͷਓछʢΧςΰϦʔʣ
தࠃਓͷͷΦοζൺ0.5952ɻ
ΧςΰϦʔͷ࣌ϕʔεϨϕϧͱൺΔɻ
தࠃਓͷനਓͷʹൺͯΦοζൺ0.5952ߴ͍ɻ
தࠃਓͷനਓͷʹൺͯΦοζൺ0.5952ߴ͍ɻ ʁʁʁ
ϐϘοτςʔϒϧΛ࡞ͬͯߟ͑ͯΈΔɻ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954
֬Λܭࢉ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 தࠃਓͷͷ࣌ʹTRUEʹͳΔ֬ʁ 296
(TRUE) / (296+3,839) (Total) = 0.072 (7.2%)
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 39,221 (TRUE)
/ (39,221+311,954) (Total) = 0.112 (11.2%) നਓͷͷ࣌ʹTRUEʹͳΔ֬ʁ
ΦοζΛܭࢉ
96 Φοζ Φοζ = TRUEͷ֬ / FALSEͷ֬
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 TRUEͷ֬: 296
/ (296+3,839) = 0.072 FALSEͷ֬: 1 - 0.072 = 0.928 Φοζ: 0.072 / 0.928 = 0.077 தࠃਓͷͷ࣌ʹTRUEʹͳΔΦοζʁ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 TRUEͷ֬: 39,221
/ (39,221 + 311,954) = 0.112 FALSEͷ֬: 1 - 0.112 = 0.888 Φοζ: 0.112 / 0.888 = 0.126 നਓͷͷ࣌ʹTRUEʹͳΔΦοζʁ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 0.126 0.077
Φοζ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 0.126 0.077
നਓʹൺͯதࠃਓ͕TRUEʹͳΔΦοζʁ Φοζ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 0.126 0.077
നਓʹൺͯதࠃਓ͕TRUEʹͳΔΦοζʁ 0.077 / 0.126 = 0.611 Φοζ Φοζൺ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 0.126 0.077
നਓʹൺͯதࠃਓ͕TRUEʹͳΔΦοζ0.611ഒʁ 0.077 / 0.126 = 0.611 Φοζ Φοζൺ
தࠃਓ നਓ TRUE 296 39,221 FALSE 3,839 311,954 0.126 0.077
നਓʹൺͯதࠃਓ͕TRUEʹͳΔΦοζ40ˋ͍ʁ 0.077 / 0.126 = 0.611 Φοζ Φοζൺ
The odds of Chinese Mothers having premature babies is 40%
less likely compared to White Mothers.
࣮͜͏͍͏දݱαΠΤϯεؔ࿈ͷ ൃදͰΑ͘Έ͔͚Δɻ
Source: More meat, more problems: Bacon may increase breast cancer
risk in Latinas. U of South Carolina News, Zen Vuong, March 3 2016 “ϕʔίϯΛຖ20άϥϜ΄Ͳ৯Δϥςϯܥͷঁੑ͕ೕ͕Μ ʹͳΔՄೳੑϕʔίϯΛ৯ͳ͍ϥςϯܥͷঁੑʹൺͯ 42ˋߴ͘ͳΔ͜ͱ͕ݚڀͷ݁ՌΘ͔ͬͨɻ”
Source: More meat, more problems: Bacon may increase breast cancer
risk in Latinas. U of South Carolina News, Zen Vuong, March 3 2016 “ϕʔίϯΛຖ20άϥϜ΄Ͳ৯Δϥςϯܥͷঁੑ͕ೕ͕Μ ʹͳΔΦοζϕʔίϯΛ৯ͳ͍ϥςϯܥͷঁੑʹൺͯ 1.42ഒͰ͋Δ͜ͱ͕ݚڀͷ݁ՌΘ͔ͬͨɻ”
108 ΦοζൺͷՄࢹԽ มͷࢦඪͱͯ͠ɺΦοζൺΛબ͢Δɻ
Odds Ratio = exp(Coefficient)
110 ͷྸ͕1ࡀ্͕Δͱɺ͕35ࡀҎ্Ͱ͋Δ Φοζ͕1.3ഒ্͕Δɻ
Φοζൺ͕Α͘ཧղग़དྷͳ͍ਓɻ ৺͠ͳ͍Ͱ͍ͩ͘͞ɻ
͏গ͠ײతͳࢦඪ͕͋Γ·͢ɻ
113 ϩδεςΟοΫճؼ • ʢCoefficientʣ • ΦοζൺʢOdds Ratioʣ • ฏۉݶքޮՌʢAverage Marginal
Effectʣ
114 ฏۉݶքޮՌ (Average Marginal Effect)
ฏۉݶքޮՌ (Average Marginal Effect) ม͕1্͕Δͱɺ͕֬ฏۉͯ͠ͲΕ্͚͕ͩΔͷ͔Λࣔ͢ɻ
ϩδεςΟοΫۂઢ
None
͋Δۃͷ͖ ݶքޮՌ
119 • ݶքޮՌɺ֤σʔλʹΑͬͯҧ͏ͷͰɺ͜ͷ·· ͰҰͭͷมͷࢦඪʹͳΒͳ͍ɻ • ͯ͢ͷσʔλʹ͍ͭͯݶքޮՌΛฏۉͯ͠Ұͭͷม ͷࢦඪʹͨ͠ͷ͕ฏۉݶքޮՌɻ ฏۉݶքޮՌ
ݶքޮՌ ͯ͢ͷσʔλͷݶքޮՌͷฏۉ
121 มͷࢦඪʹฏۉݶքޮՌΛબͿɻʢσϑΥϧτʣ
122 ฏۉݶքޮՌ ͷྸ͕1ࡀ্͕Δͱɺ͕̏̑ࡀҎ্ Ͱ͋Δ͕֬ฏۉͯ͠3%΄Ͳ͕͋Δɻ
มͷӨڹʹؔ͢Δ౷ܭςετ ʢԾઆݕఆʣ
PʢP Valueʣ
125 • ؼແԾઆɺʮ͜ͷมɺ࣮༧ଌ͍ͨ͠ͱؔͳ͍ɻʢͦ͏Έ ͑ΔͷۮવͰ͋Δʣʯ • P ɺؼແԾઆ͕ͳΓͨͭͱͨ͠ͱ͖ʹɺ࣮ࡍʹग़͍ͯΔͱಉఔ ͔ͦΕҎ্ʹมͱ݁Ռ͕ؔ࿈͍ͯ͠ΔΑ͏ʹݟ͑Δ֬ɻ • P͕
5%ҎԼͰ͋ΕɺؼແԾઆغ٫ग़དྷΔͷͰɺม݁Ռͱؔ ࿈͕͋Δͱߟ͑Δɻ PʢP Valueʣ
126 ༧ଌม͕1͚ͭͩͷ߹ΛΈ͖ͯͨɻ
Simple Logistic Regression P(y) = logistic(a * x + b)
ͪΌΜͷ͕1૿͑Δͱɺૣ࢈ʹͳΔΦοζ͕ 13ഒʹͳΔɻ Φοζൺͷ߹
ͪΌΜͷ͕1૿͑Δͱɺૣ࢈ʹͳΔ͕֬ฏۉͰ 23.67%্͕Δɻ ฏۉݶքޮՌͷ߹
ฏۉݶքޮՌͷ߹
131 ༧ଌม͕ෳͷ߹ɻ
Multiple Logistic Regression P(y) = logistic(a1 * x1 + a2
* x2 + b)
ෳͷྻΛ༧ଌมͱͯ͠બͿɻ
ଞͷมͷ͕ҰఆͰ͋Εɺ ͪΌΜͷ͕૿͑Δͱૣ࢈ʹͳΔΦοζ2.68ഒʹͳΔɻ Φοζൺͷ߹
ଞͷมͷ͕ҰఆͰ͋Εɺ ͪΌΜͷ͕૿͑Δͱૣ࢈ʹͳΔ֬ฏۉͰ7ˋ্͕Δɻ ฏۉݶքޮՌͷ߹
ฏۉݶքޮՌͷ߹
Q & A
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࿈བྷઌ ϝʔϧ
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