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比較百分比 (percetage) 卡方檢定 (Ch-square tests) 比較百分比觀察一群人 () 中, 產生某一現象之人數 () 盛行率 (p= ) 或是百分比 ( X00%) 盛行率 p 與另一固定盛行率 p 0 比較, 做一單樣本的檢定 : 百分比的單樣本檢定, 是指盛行率 p 與一固定數 p 0 比較, 是否有 pˆ p0 顯著性差異其檢定公式 z s =, 其餘步驟可參考 Z 檢定 ; p ( p ) 檢定公式中,p 是代表母全體的盛行率 pˆ, 則是代表由樣本所計算出來的盛行率 0 0 資料中分成兩組後比較此兩組盛行率的差別 (p - P), 做一雙樣本的比較如果是資料中分成兩組後, 比較此兩組盛行率的差別 (p -p ), pˆ ˆ p z 其檢定公式為 s = pˆ + pˆ 呈常態分配, 其中 pˆ = pˆ qˆ ( + ) + 其餘檢定步驟亦可參考 Z 檢定, 需特別注意的 qˆ = pˆ 是使用這種方法時必須 pq ˆ ˆ 5 且 pq ˆ ˆ 5才適用

卡方檢定 Desg 探討兩類別變項 (categorcal varables) 是否相關 Outcome (y) Cotuous Cotuous Pearso Correlato coeff. Lear regresso Spearma Correlato coeff. Categorcal t-test / AOVA Lear models Wlcoxo Rak-Sum test Kruskal-Walls test Categorcal Logstc Categorcal aalyss χ Fsher s Exact Categorcal aalyss Logstc aalyss 卡方檢定假設變項 A 有 r 個選項, 變項 B 有 c 個選項, 那麼這兩個變項的資料就可以被整理成變項 A ( 有 r 個選項 ) 每人僅可單選 r 變項 B( 有 c 個選項 ) 每人僅可單選 3 c r r 3 3 r3 c c rc H 0 : 變項 A 及變項 B 互相獨立 H : 變項 A 及變項 B 沒有互相獨立或 H 0 : 變項 A 及變項 B 沒有關係 H : 變項 A 及變項 B 有關係

原始資料所反映之觀察值 : 計算出之期望值 : 變項 B 變項 A 3 3 3......... c c c............ c. c. r r r r3... rc r....3....c r.. r.. r.. c.. j 由 O j (= j ) 代表, 期望值由 Ej(= r c 卡方檢定值為 χ = ( O j E j ) s 呈自由度 (r-)(c-) 的卡方分配 χ (r-)(c-), 臨界值為 χ (r-)(c-)-α, = j = P- 值為為 Pr(χ (r-)(c-) >χ ) E j ) 代表, 則 < 舉例 > 假設要探討職位分類是否與贊成週休二日有關, 經過調查後可整理成以下的表格 : 是否贊成週休日職位分類職員教員主管贊成 30 40 0 反對 5 50 5 未決定 5 0 5 合計 60 00 40 合計 80 90 30 00

因此虛無假設與對立假設之寫法為 H 0 : 對週休二日的意見與職位分類無關 H : 對週休二日的意見與職位分類有關根據上面表格計算出期望值 : 是否贊成週休日職位分類職員教員主管 60 80 00 00 80 00 40 80 00 贊成 =4 =40 =6 60 90 00 00 90 00 40 90 00 反對 =7 =45 =8 60 30 00 00 30 00 40 30 00 未決定 =9 =5 =6 合計因此假設檢定值為 (30 4) (5 7) (5 6) x s = + + + = 8.94444 4 7 00 6 自由度為 df = (3-) x (3-)=4, 因此臨界值為 χ 4.095=9.49, 因為 8.9 >9.49 所以推翻 H 0 ; 檢定結果顯示職位與是否贊成週休二日有顯著性相關針對 r x c table 使用 χ test 之注意事項 Page 396,use ths test oly f the followg two Codtos are satsfed: (a) o more tha 0% of the cells should have expected values 5. (b) o cell should be expected value. 一般常用於檢定 x tables 者有 χ test χ test wth Yates correcto Fsher s exact test

. 一般而言 r 或 c 的項目不可過多. Ej 5 時 χ test 所給的 p-value 才會較正確 3. 當 j 或 Ej 過小時, 可考慮 combe categores 4. Fsher s exact test for RxC tables 5. Ch-square testg for tred Ch-square testg for tred 檢定個 categorcal varables 之 assocato 時, χ -test 是基本方法若個變數都是 omal 則僅可用 χ test(df=(r-)(c-)) 若其中個或個是 ordal 則可用更好 (powerful) 的方法 (df 較小的方法 ) 所謂更好的方法是找出適當的分數目來取代 ordal varables 例如 : VAR AGECAT 項目 3 4 5 意義 0-0 0-30 30-40 40-50 50 分數 (score) 0 5 35 45 70 亦可用其他種分數取代例如 : 項目非常滿意滿意中立不滿意非常不滿意滿意度分數 3 4 5 人數 30 40 50 0 0 Stadardzed mdrak score 0.06 0.3344 0.635 0.864 0.9636 以整數來當分數者, 僅 cotuous varable 轉換而來之 categorcal varable 才適合. 若僅是程度上的差異則建議用 stadardzed mdrak score. 相當於執行無母數分析 (Y: 滿意度分數 )

Ch-square tests X reflex % Y o-reflex % total Lght-eyed 54 49.0% 564 5.0% 06 如何做表? Dark-eyed total 3 854 47.7% 48.5% 34 906 5.3% 5.5% 654 760 χ df =0.8, p=0.6000 Commuty survey Whether X geotype related to oral cacer/precacer? Sample presetato: From the sample of 760 patets, 54 of the 06 (49.0%) lght-eyed partcpats ad 3 of 654 (47.7%) dark-eyed partcpats exhbted the reflex respose. The ch-square test revealed that reflex respose ad eye color were ot statstcally sgfcatly assocated( χ df=0.8, p=0.6000). From: Lag & Secc, How to report statstcs medce. d (006) % horzota lly sum up to 00% Cacer PreCacer ormal p-value of total % % % ch-square total 3 04 48.83 9.86 88 4.3 GG 60 30 50.00 8 3.33 36.67 0.4595 TG 98 5 5.04 9 9.8 38 38.78 TT 55 3 4.8 4 7.7 8 50.9 Subjects wth TT geotype had hgher percetage free of dseases (cacer or precacer) (50.9%) the GG (36.67%) ad TG (38.78%). The dfferece (or assocato) was ot statstcally sgfcat (p=0.4595).

Case-cotrol study Whether X geotype related to oral cacer/precacer? X vs Y 之交叉分析 % vertcally sum up to 00% Cacer PreCacer ormal p-value of total % % % ch-square total 3 04 88 GG 60 30 8.85 8 38.0 5.00 0.4595 TG 98 5 49.04 9 4.86 38 43.8 TT 55 3.4 4 9.05 8 3.8 ormal subjects had hgher percetage of TT (3.8%) tha cacer (.4%) or precacer (9.05%) patets. The dfferece (or assocato) was ot statstcally sgfcat (p=0.4595). Chug et al, Brtsh Joural of Cacer (005) 93:60-606 範例一 : 各種白斑症病理組織分析結果範例二 : 瑞典扁平苔癬的型態分佈均質性白斑症沒有上皮變異個案數 73 百分比 (%) 77.7 輕微或中等的上皮變異個案數百分比 (%).3 嚴重的上皮變異與原位癌及上皮癌個案數 0 百分比 (%) 0.0 合計 94 型態 ( 依總合之百分比排序 ) 病患人數 () 男病患中所佔之比例 (%) 病患人數 () 女病患中所佔之比例 (%) 病患人數 () 總和病患中所佔之比例 (%) 疣狀白斑症 6 33.3 7 38.9 5 7.8 8 合計 49 453 70 紅白斑症結節狀白斑症合計 87 33.3 60.8 X =54.5, df=6, p<0.00 4 4. 0 40 5.6 6.7 8.0 5 6 6 6.3 50.0 4. 9 43 網狀斑狀萎縮狀丘疹狀潰瘍狀皰狀 67 4 9 8 45.0 6.9 6.9 7.6 3. 0.4 05 0 9 0 4 45.3 4.3 0.3 4.4 5.3 0.4 37 77 34 39 3 3 45. 5. 9. 5.6 4.6 0.4 χ =5.97, df=5, p=0.3094

Measures of Effect x Tables Ch-square tests Odds Ratos ad Mattel-Haeszel Test Dsease vs Exposure Pe:Exposed 者之得病率 Pue:U-exposed 者之得病率 Pe Rsk Rato (RR)= Pue Pe Odds of exposure= ( Pe) Odds of u-exposure= Odds rato (OR)= Pe /( Pe) Pue /( Pue) Pue ( Pue) 注意 : 當 Pe&Pue 很小時, (-Pe), (-Pue) Pe /( Pe) Pue /( Pue) 則 OR= Pe = RR Pue Measure of effect sze cross tab Odds ratos (OR) = (a x d)/(b x c) Dsease o dsease exposure a b o exposure c d

Compute Odds Rato (crude) Compute Odds Rato (crude) Cacer/ormal PreCacer/ormal Cacer PreCacer ormal (Crude) (Crude) OR OR GG 30 8.66.55 TG 5 9 38.63.66 TT 3 4 8.00.00 Cacer/ormal PreCacer/ormal Cacer PreCacer ormal (Crude) (Crude) OR OR GG a b c (a x m)/(c x g) (b x m)/(c x h) TG d e f (d x m)/(f x g) (e x m)/(f x h) TT g h m.00.00 PreCacer/ormal (Crude) OR ( 95%CI ) p-value GG.55 ( 0.68, 9.57 ) 0.666 TG.66 ( 0.46, 5.93 ) 0.437 TT.00 Comparg to geotype TT, people wth GG had.55 tmes (95% CI=.68, 9.57) of the chace for havg precacer. 95% cofdece tervals Cofouders & Stratfcato 干擾因素與分類因素 Cofdece tervals s mportat for people to see the effcecy Ca blow up by small cell sze Dr Dr LC 0.094 0.07 LC BUT! Smokers o-smokers LC LC LC LC Dr 0.03 Dr 0.0 Dr 0.03 Dr 0.0

要去掉干擾或分類因素的影響, 可用 Matel- Haeszel Test 做法是將原來的一個表格, 相關係數 & 廻歸係數依因素分成多個表再針對每一個表算出 χ test statstcs 之後再整合起來 Correlato Coeffcet & Regresso coeffcet Chapter Sectos.-.8 相關係數 (Correlato Coeffcet) Desg Outcome (y) Cotuous Categorcal Cotuous Pearso Correlato coeff. Lear regresso Spearma Correlato coeff. Logstc Categorcal aalyss Categorcal t-test / AOVA Lear models Wlcoxo Rak-Sum test Kruskal-Walls test χ Fsher s Exact Categorcal aalyss Logstc aalyss 要了解兩個數值變項 ( 等距尺度等比尺度 ) 之相關性時, 可以利用皮爾森相關係數 (Pearso Correlato Coeffcet) 來探討, 其中母全體的真值以 ρ 來代表皮爾森相關係數主要是測量兩變數間之線性 (lear) 關係, 因此兩變項間是具有曲線關係時, 皮爾森相關係數則無法測量針對變項 x 與變項 y 之皮爾森相關係數的樣本值 r

計算公式為 : r = [ = ( x x)( y y) = ( x x) ][ = ( y y) ] 若已知 Σx, Σx, Σy, Σy, Σx y I 則較簡化的計算公式為 r = y [ ( x )( y ) x y, ( x ) ( y ) x ][ y, S x, S y,σx y I x 若已知, y,, S x, S y,σx y I 之計算公式相關係數是一個 -~ 的數字, 正值表正相關, 負值表負相關, 零表沒有相關, 離零越遠則相關性越強一般來說, 若相關係數大於 0.75 則可視為非常相關,0.5~0.75 則為普遍相關 x y xy r = ( ) S S x y Hypothess Testg for ρ ㄧ般統計軟體提供的 p-value 是當 x y 呈 ormal dstrbuto 要檢定 ρ 是否不同於 0 時當 x y 不一定呈 ormal dstrbuto 要檢定 ρ 是否不同於 ρ 0 時, 先用 Fsher Z Trasformato of the r, 再作檢定 Pearso Correlato Coeffcet 常被用來當作 regresso 之前置步驟 Page 07, Pearso ad Turto: Statstcal methods evrometal health

相關係數的表示 Sample Presetato: Detee lead levels correlated well ad versely wth famly come, dcatg that poorer chldre have hgher levels of lead ther systems (=39; Pearso s r=- 0.6; P=0.00). Page 08, Pearso ad Turto: Statstcal methods evrometal health From: Lag & Secc, How to report statstcs medce. d (006) Orgal table Age Weght Oxy Rutme RuPulse RstPulse MaxPulse Age.0000-0.405-0.38 0.95-0.36-0.509-0.449 Weght -0.405.0000-0.68 0.435 0.85 0.0440 0.494 Oxy -0.38-0.68.0000-0.86-0.3980-0.3994-0.367 Rutme 0.95 0.435-0.86.0000 0.336 0.4504 0.6 RuPulse -0.36 0.85-0.3980 0.336.0000 0.355 0.998 RstPulse -0.509 0.0440-0.3994 0.4504 0.355.0000 0.305 MaxPulse -0.449 0.494-0.367 0.6 0.998 0.305.0000 From: Lag & Secc, How to report statstcs medce. d (006)

dgts sortg Age Weght Oxy Rutme RuPulse RstPulse MaxPulse Age.00-0.4-0.3 0.0-0.3-0.5-0.4 Weght -0.4.00-0.6 0.4 0.8 0.04 0.5 Oxy -0.3-0.6.00-0.86-0.40-0.40-0.4 Rutme 0.0 0.4-0.86.00 0.3 0.45 0.3 RuPulse -0.3 0.8-0.40 0.3.00 0.35 0.93 RstPulse -0.5 0.04-0.40 0.45 0.35.00 0.3 MaxPulse -0.4 0.5-0.4 0.3 0.93 0.3.00 RuPulse MaxPulse Oxy Rutme RstPulse Weght Age RuPulse.00 0.93-0.40 0.3 0.35 0.8-0.3 MaxPulse 0.93.00-0.4 0.3 0.3 0.5-0.4 Oxy -0.40-0.4.00-0.86-0.40-0.6-0.3 Rutme 0.3 0.3-0.86.00 0.45 0.4 0.0 RstPulse 0.35 0.3-0.40 0.45.00 0.04-0.5 Weght 0.8 0.5-0.6 0.4 0.04.00-0.4 Age -0.3-0.4-0.3 0.0-0.5-0.4.00 Par-wse Varable by Varable Cout Correlato p-value Weght Age 3-0.405 0.95 Rutme Age 3 0.95 0.96 Rutme Weght 3 0.435 0.44 RuPulse Age 3-0.36 0.083 RuPulse Weght 3 0.85 0.384 RuPulse Rutme 3 0.336 0.0858 RuPulse MaxPulse Oxy Rutme RstPulse Weght Age RuPulse.00 0.93-0.40 0.3 0.35 0.8-0.3 MaxPulse 0.93.00-0.4 0.3 0.3 0.5-0.4 Oxy -0.40-0.4.00-0.86-0.40-0.6-0.3 Rutme 0.3 0.3-0.86.00 0.45 0.4 0.0 RstPulse 0.35 0.3-0.40 0.45.00 0.04-0.5 Weght 0.8 0.5-0.6 0.4 0.04.00-0.4 Age -0.3-0.4-0.3 0.0-0.5-0.4.00 RstPulse Age 3-0.509 0.478 RstPulse Weght 3 0.0440 0.843 RstPulse Rutme 3 0.4504 0.00 RstPulse RuPulse 3 0.355 0.058 MaxPulse Age 3-0.449 0.003 MaxPulse Weght 3 0.494 0.76 MaxPulse Rutme 3 0.6 0.3 MaxPulse RuPulse 3 0.998 <.000 MaxPulse RstPulse 3 0.305 0.095 Oxy Age 3-0.38 0.0878 Oxy Weght 3-0.68 0.387 Oxy Rutme 3-0.86 <.000 Oxy RuPulse 3-0.3980 0.066 Oxy RstPulse 3-0.3994 0.060 Oxy MaxPulse 3-0.367 0.997

Par-wse ad sorted by r Varable by Varable Cout Correlato p-value MaxPulse RuPulse 3 0.998 <.000 RstPulse Rutme 3 0.4504 0.00 RstPulse RuPulse 3 0.355 0.058 RuPulse Rutme 3 0.336 0.0858 MaxPulse RstPulse 3 0.305 0.095 MaxPulse Weght 3 0.494 0.76 MaxPulse Rutme 3 0.6 0.3 Rutme Age 3 0.95 0.96 RuPulse Weght 3 0.85 0.384 Rutme Weght 3 0.435 0.44 RstPulse Weght 3 0.0440 0.843 RstPulse Age 3-0.509 0.478 Oxy Weght 3-0.68 0.387 Oxy MaxPulse 3-0.367 0.997 Weght Age 3-0.405 0.95 Oxy Age 3-0.38 0.0878 RuPulse Age 3-0.36 0.083 Oxy RuPulse 3-0.3980 0.066 Oxy RstPulse 3-0.3994 0.060 MaxPulse Age 3-0.449 0.003 Oxy Rutme 3-0.86 <.000 簡單線性迴歸 Smple Lear Regresso y=α+βx+e y depedet varable,the value of the respose varable to be predcted x depedet varable,the explaatory varable used to predct the value of y α the pot at whch the regresso le crosses the y axs(the y tercept pot) βthe slope of the regresso le e s ormally dstrbuted wth mea 0 ad varace σ ( 注意! 迴歸分析的 ormal assumpto 是在 e 不是在 y) Page 06, Pearso ad Turto: Statstcal methods evrometal health

Estmato of the Least-Squares Le y=α+βx+e 其中 α&β 可由統計軟體中計算出來 R the proporto of the varace of y that ca be explaed by the varable x 在 smple lear regresso 中 square root of R 就是 Pearso correlato coeffcet Cofdece tervals of parameters Cofdece tervals for predcto Sample Presetato: From our 453 partcpats,we attempted to predct serum levels from weght usg smple lear regresso aalyss. The slope of the regresso le was sgfcatly greater tha zero,dcatg that serum level teds to crease as weght creases (slope=0.5; 95%CI=0.9 to 0.3;t 45 =8.3; p<0.00; y=.6+0.5x;r =0.67). From: Lag & Secc, How to report statstcs medce. d (006) Dfferece betwee Correlato coeffcet (r) & Regresso Coeffcet (β) r = r β = β

r > r β = β r = r β > β r < r β > β Correlato Regresso Varable by Varable Cout r p-value β p-value Oxy Rutme 3-0.86 <.000-3.3 <.000 Oxy RstPulse 3-0.40 0.060-0.8 0.060 Oxy RuPulse 3-0.40 0.066-0. 0.066 Oxy Age 3-0.3 0.0878-0.3 0.0878 Oxy MaxPulse 3-0.4 0.997-0.4 0.997 Oxy Weght 3-0.6 0.387-0.0 0.387

Ay questos? 引用圖文出處 : Roser: Fudametals of Bostatstcs, 6th. Wadsworth Publshg Compay. 公共衛生學 : 4th ed., 邱清華總校閱, 華杏出版社. Lag & Secc: How to report statstcs medce. d ed (006) Peraso & Turto: Statstcal methods evrometal health. Chapma ad Hall