悪い経時データ解析例の解説

(1)

悪い経時データ解析例の解説

浜田知久馬

9/25(金)

EUA研究会

(2)

発表構成

2

・

Journal of Pharmacological Scienceの

経時データ解析例の紹介

・輪切りの解析を行った場合のαエラー

・不適切な経時型分散分析

個体差を無視した解析

(3)

要約

非臨床の薬理データでは，しばしば経時測定

データについての解析が行われる．この解析

方法として様々な方法が提案されているが，

薬理学研究では不適切な方法が用いられる

ことも多い．

本発表ではJournal of Pharmacological Science

の文献調査の結果から不適切な経時データ

の統計解析と記載例について，不適切な理

由を解説する．具体的には，時点の輪切りの

解析で多重性が考慮されてない，個体間の

対応が無視されている，分散分析表が記載さ

れてない等の問題がみられた．

(4)

(5)

薬理学研究での統計手法の実態

典型的な誤用とその解決方法

浜田知久馬

西沢友恵

日本薬理学雑誌，133‐6,2‐9(2009)

EUAでは分散分析のみ報告

(6)

6

分散分析

ＡＮＯＶＡ：

_ＡN

ａｌｙｉｓｉｓ

_O

_f

_VA

_riance

分散(variance)：ばらつきを表す統計用語

分散分析：データのばらつきを構成する要因が複数

存在するときに個々の要素に分解

二元配置(two-way)分散分析

要因の

例）

薬剤（Ａ，Ｂ，Ｃ）

用量(6,7,8)

(7)

7

A

B

C

薬剤

用量

(8)

8

用語

分散分析と共分散分析

ＡＮＯＶＡ：

_{ＡNａｌｙｉｓｉｓ}

_{Of VAriance}

分散分析（群間比較）

ＡＮCＯＶＡ：

ＡN

ａｌｙｉｓｉｓ

_of

_COVA

_riance

共分散分析

反応変数に相関のある変数COを考慮し

た分散分析ANOVA（群間比較）

CO：

covarite (共変量）

(9)

9

調査対象文献

• JPHS : Journal of Pharmacological

Science 2007年の論文133報.

• JJP : The Japanese Journal of

Pharmacology（現在のJournal of

Pharmacological Science)

2002年の論文149報.

1996年の論文134報.

(10)

10

統計手法

(分散分析）

1996年 (n=134) 2002年 (n=149) 2007年 (n=133) n % n % n % 1 way ANOVA 46 34.3 75 50.3 74 55.6 2 way ANOVA 6 4.5 11 7.4 11 8.3 Kruskal-Wallis test 8 6.0 3 2.0 5 3.8 Freidman test 0 0.0 0 0.0 1 0.8

Repeated measures 1 way ANOVA

3 2.2

10 6.7 4 3.0

Repeated measures 2 way

ANOVA 2 1.3 8 6.0

Repeated measures MANOVA 0 0.0 1 0.7 0 0.0

結果

(11)

経時データの特徴

１）

同一対象者内の測定値は他の対象者の

測定値よりも似ている

（独立性が成り立たない）

2）同一対象者内でも近い時点で測定された

値は離れたものに似ている

（系列相関）

３）投与前値の存在

４）多時点で検定（多重性）

11

(12)

連続型反応変数の

分散共分散行列

12

1

2

3

4 対象者

X

2

4

6

8 Y

10

7

12

11

2 σ

0

(13)

経時データ

13

1

2

3

4 対象者

X

2

4

6

8 Y

_t=1

10

7

12

11 Y

_t=2

6

5

9

14 Y

_t=3

6

3

8

16

(14)

経時データにおける

分散共分散行列

14 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 2 2 2 3 3 3 4 4 4 対象者時点 4 4 4 2 2 2 6 6 6 8 8 8 10 6 6 7 5 3 12 9 8 11 14 16 X Y 1

V

2

V

3

V

4

V

0

(15)

15

経時データの解析例

7.J Pharmacol Sci 105, 66

–

73 (2007)

Impact of Systemic Histamine Deficiency on the Impact of Systemic Histamine Deficiency on the

Crosstalk Between Mammary Adenocarcinoma

and T Cells

Immunofluorescence labeling and flow cytometryImmunofluorescence labeling and flow cytometry

Lymphocytes were isolated from the spleen and were

used for two

used for two- - and threeand three--color cell surface labeling using color cell surface labeling using Abs against CD3, CD8, CD4, and CD25. All Abs were

Abs against CD3, CD8, CD4, and CD25. All Abs were

purchased from BD PharMingen (BD Biosciences,

Franklin Lakes, NJ, USA). Labeled cells were analyzed by

using FACSCalibur flow cytometer and CellQuest

software (BD Biosciences). For each sample, an

isotypecontrol was used to determine the positive and

negative cell populations. The P values were determined

by

(16)

16 16 経時的データに輪切りのｔ検定の繰り返し

（時点

10 群2）

後値

(17)

17

52.

52. J Pharmacol Sci 104, 130

J Pharmacol Sci 104, 130

–

136 (2007)

Statistical analysesStatistical analyses

Temperature response was assessed as changes from

pre

pre--injection values (injection values (Δ°Δ°C) or the fever index (FI), the C) or the fever index (FI), the area under the curve produced in the 5

area under the curve produced in the 5--h period after h period after the injection of LPS, in term of degrees centigrade per 5

the injection of LPS, in term of degrees centigrade per 5

h was calculated (8). The 2,3

h was calculated (8). The 2,3--DHBA level of samples was DHBA level of samples was expressed as a percent of the mean baseline. In addition,

expressed as a percent of the mean baseline. In addition,

results about body temperatures, the fever index, and

levels of PGE2 are expressed as the mean

levels of PGE2 are expressed as the mean ±± S.E.M. for S.E.M. for n experiments.

n experiments. Two way analysis of variance (ANOVA) Two way analysis of variance (ANOVA) for repeated measurements

for repeated measurements (in the same animals) was (in the same animals) was used for the factorial experiment, whereas

used for the factorial experiment, whereas DuncanDuncan’s’s test test was used for post hoc multiple comparisons among

was used for post hoc multiple comparisons among

means. A P value less than 0.05 was considered to

indicate a statistically significant difference.

(18)

18

経時データの輪切りの検定にDuncan

（時点

15 群5）

(19)

19

118.

118. J Pharmacol Sci 103, 293

J Pharmacol Sci 103, 293

–

298 (2007)

Statistical analysisStatistical analysis

Temperature response was assessed as changes from

pre

pre--injection values (injection values (Δ°Δ°C) and the fever index (FI), C) and the fever index (FI), which was the area under the curve produced in the 5

which was the area under the curve produced in the 5--h h period after the injection of LPS in term of degrees

period after the injection of LPS in term of degrees

centigrade per 5 h, were calculated (1, 2). The

glutamate and 2,3

glutamate and 2,3--DHBA levels of samples were DHBA levels of samples were

expressed as a percentage of three consecutive mean

baseline values. Results were expressed as the mean

baseline values. Results were expressed as the mean ±± S.E.M. for n experiments.

S.E.M. for n experiments. Two way analysis of variance Two way analysis of variance (ANOVA) for repeated measurements

(ANOVA) for repeated measurements (in the same (in the same

animals) was used for the factorial experiment, whereas

Dunnett

Dunnett’’s test was used for post hoc multiple s test was used for post hoc multiple comparisons

comparisons among means. A P value less than 0.05 among means. A P value less than 0.05 was considered to indicate a statistically significant

was considered to indicate a statistically significant

difference.

(20)

20

経時データの輪切り検定にDunnett

(21)

21

54.

54. J Pharmacol Sci 104, 146

J Pharmacol Sci 104, 146

–

152 (2007)

Statistical analysesStatistical analyses

Experimental data were presented as the mean

Experimental data were presented as the mean ±±S.E.M. S.E.M. The effects of CLE on the body weight and the weights

The effects of CLE on the body weight and the weights

of visceral organs were tested by a

of visceral organs were tested by a oneone--way analysis of way analysis of variance (ANOVA).

variance (ANOVA). The effects of CLE on the number of The effects of CLE on the number of RBCs, hemoglobin concentration, hematocrit, MCV, MCH,

RBCs, hemoglobin concentration, hematocrit, MCV, MCH,

MCHC, and the number of WBCs were evaluated by a

two

two--way ANOVA for repeated measuresway ANOVA for repeated measures. Subsequent . Subsequent post hoc analyses to determine significant differences

post hoc analyses to determine significant differences

between two groups and from day 0 in each group were

performed by

performed by FisherFisher’’s protected least significant s protected least significant difference (PLSD) test and Dunnett

difference (PLSD) test and Dunnett’’s test, s test, respectively. respectively. The differences were considered significant when P was

The differences were considered significant when P was

<0.05.

(22)

22 22 経時データの輪切りの解析にｔ検定時点0との比較にDunnett検定

（時点４

群２）

後値

(23)

23

63.

63. J Pharmacol Sci 104, 212

J Pharmacol Sci 104, 212

–

217 (2007)

NG

--NitroNitro--

L

--arginine Methyl Ester, but Not arginine Methyl Ester, but Not

Methylene Blue, Attenuates Anaphylactic

Hypotension in Anesthetized Mice

StatisticsStatistics

All results are expressed as the means

All results are expressed as the means ±± S.D. Statistical S.D. Statistical analysis was performed by

analysis was performed by repeated measures analysis repeated measures analysis of variance.

of variance. Comparison of individual points within Comparison of individual points within groups was made by

groups was made by analysis of variance followed by the analysis of variance followed by the Bonferroni post

Bonferroni post--test correction method. test correction method. Comparison of Comparison of individual points between two groups and among four

individual points between two groups and among four

groups was made by

groups was made by StudentStudent’’s ts t--test and analysis of test and analysis of variance followed by the Bonferroni post

variance followed by the Bonferroni post--test correction test correction method

method, respectively. Differences were considered , respectively. Differences were considered statistically

statistically significant at P<0.05.significant at P<0.05.

経時データの輪切りの検定

(24)

24 24 経時データの輪切りの検定にBonferroni *：投与前との比較 #:controlとの比較

（時点

20 群４）

差

(25)

25

4.J Pharmacol Sci 105, 41

–

47 (2007)

Possible Involvement of 5

-

Lipoxygenase

Metabolite in Itch

-

Associated Response of

Mosquito Allergy in Mice

Statistical analysisStatistical analysis

Values given are means and S.E.M. Statistical

significance was analyzed using

significance was analyzed using StudentStudent’’s ts t--test, test, Dunnett

Dunnett’’s multiple comparisons, or twos multiple comparisons, or two--way repeated way repeated measures analysis of variance

measures analysis of variance; P<0.05 was considered ; P<0.05 was considered significant.

(26)

26

経時型分散分析の結果を表示

（時点

20 群3）

(27)

27

64.

64. J Pharmacol Sci 104, 218

J Pharmacol Sci 104, 218

–

224 (2007)

Effects of Loperamide on Mechanical Allodynia Induced by Effects of Loperamide on Mechanical Allodynia Induced by

Herpes Simplex Virus Type

Herpes Simplex Virus Type--1 in Mice1 in Mice

Data analysesData analyses

The means of data are presented together with S.E.M.

Data on the time course of anti

Data on the time course of anti--allodynic effects were allodynic effects were analyzed with

analyzed with Friedman repeated measures analysis of Friedman repeated measures analysis of variance

variance on ranks and then with Dunnetton ranks and then with Dunnett’’s test. s test. Other Other data were analyzed with

data were analyzed with the the Mann-Mann-Whitney rank sum Whitney rank sum test or Wilcoxon signed rank test.

test or Wilcoxon signed rank test. P<0.05 was P<0.05 was considered significant.

considered significant.

ノンパラメトリックな経時データの解析

(28)

28

経時データを順位に変換してDunnett

（時点３

群３）

(29)

29

111.

111. J Pharmacol Sci 103, 234

J Pharmacol Sci 103, 234

–

240 (2007)

Effect of a Dihydrobenzofuran Derivative

on Lipid Hydroperoxide

-

Induced Rabbit

Corneal Neovascularization

Statistical analysis

Data were compared between groups by

multiple comparisons (

William

’

s

) and these were

expressed as a mean

±

S.E.M. The differences

were considered significant when P<0.05.

(30)

30

経時データの輪切り検定にWilliams

（時点５

群３）

(31)

31

58.

58. J Pharmacol Sci 104, 176

J Pharmacol Sci 104, 176

–

182 (2007)

Effects of Combined Oleoyl

Effects of Combined Oleoyl--Estrone and Rimonabant Estrone and Rimonabant on Overweight Rats

on Overweight Rats

Statistical comparisons between groups were

established by

two

-

way ANOVAs

.

The data are the mean

±

S.E.M.

Significance of the differences between groups

(ANOVA): P values for the overall effects of OE

(P OE) and rimonabant (P RI).

(32)

32 32 三元配置経時データで最終時点で二元配置で解析交互作用を検討すべき

（時点

10 群4）

比

(33)

33

Common Mistakes

1)

分散分析のモデル，結果が示されてない．

2)

経時データの多重性

多群

×

多時点

通常の

₁

_-

_way

用の多重比較法では多重性の

調整はできない

３）投与前値の扱いについて明示的な記載がな

いことが多く，また投与前値が適切に利用さ

れてない．

(34)

経時データの多重性の対処

解析時点の限定

複数時点の要約統計量の算出

経時型分散分析で有意なときのみ群間比較

時点間比較（閉手順）

1

1 -

-

way

：時間が有意比較

2

2 -

-

way

：群有意，群

×

時点ｎｓ→群間比較

群

×

時点有意→時点の輪切り検定

多重比較（

Bonferroni

_Bonferroni

法等）

34 34

(35)

経時データの分類

一元配置

(1

-

way)

経時データ

要因：時点

誤差：個体間，個体内

二元配置

(2

-

way)

経時データ

要因：群，時点

誤差：個体間，個体内

_{誤差：個体間，個体内}

三元配置

(3

-

way)

経時データ

要因：群，用量，時点

誤差：個体間，個体内

35

(36)

経時解析のαエラー

１）完全帰無仮説の下で，ある基準群に対して，他

の３群とDunnett型の平均値の比較を両側5%の

ｔ検定で行うときの，αエラーの確率を求めよ．

２）１）の解析を，５時点について輪切りで検定を行

うときの完全帰無仮説を示せ．

３）

５時点について輪切りでｔ検定を，計５×３＝

１５回を行うときのαエラーの確率を求めよ．

４）群の数を2～8，時点の数を2‐20に変化させて

αエラーの確率を求めよ．

36

(37)

完全帰無仮説

５時点４群のDunnett型比較

54 53 52 51 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

,

:

μ

=

mean

population

j

group

i

time

ij

(38)

data data;

cvalue=probit(1‐0.05/2);

array y(4) y1‐y4;

array t(5) t1‐t5;

do j=1 to 100000;

do k=1 to 5;

do i=1 to 4;y(i)=rannor(4989);end;

t12=abs(y2‐y1)/sqrt(2);t13=abs(y3‐y1)/sqrt(2);

t14=abs(y4‐y1)/sqrt(2);

t(k)=max(t12,t13,t14);s='ns';

maxt=max(of t1‐t5);end;

if maxt ge cvalue then s='* ';

output;

end;

proc freq;tables s;run;

38

(39)

αエラーの確率は48.8%

(時点5

群4)

data data;

z=probit(1‐0.025);

p=1‐probmc('dunnett2',z,.,.,3);

alpha=1‐(1‐p)5;**

proc print;run;

(40)

時点1～20

群2～8

data data;

do g=2 to 8;

do t=1 to 20;

z=probit(1‐0.025);

p=1‐probmc('dunnett2',z,.,.,g‐1);

alpha=1‐(1‐p)**t;

output;

end;end;

proc transpose;var alpha;id t;by g;

proc print;run;

40

(41)

αエラーの確率(時点1～７

群2～8)

時点数

群数

1

2

3

4

5

6

7 2 0.050 0.098 0.143

0.185 0.226 0.265 0.302

3 0.091 0.173

0.248 0.317 0.379 0.435 0.486

4 0.125 0.235 0.331 0.415 0.488 0.553 0.609

5 0.156 0.287

0.398

0.492

0.571 0.638 0.694

6 0.183 0.332 0.454 0.554 0.636 0.702 0.757

7 0.207 0.372 0.502 0.605 0.687 0.752 0.803

8 0.230 0.406 0.543 0.648 0.729 0.791 0.839

(42)

αエラーの確率(時点8～14

群2～8)

時点数

群数

8

9

10

11

12

13

14

2 0.337 0.370

0.401 0.431 0.460 0.487 0.512

3 0.533 0.575 0.614 0.649 0.681 0.710 0.736

4 0.658 0.701

0.738 0.771 0.800 0.825 0.847

5 0.742 0.782 0.816 0.845 0.869 0.889 0.907

6 0.801 0.838 0.867 0.892 0.911 0.928 0.941

7 0.844 0.876 0.902 0.922 0.938 0.951 0.961

8 0.876 0.904 0.926 0.943 0.956 0.966 0.974

42

(43)

αエラーの確率(時点15～20

群2～

8)

時点数

群数

15

16

17

18

19

20 2 0.537 0.560 0.582 0.603

0.623

0.642 3 0.760 0.782 0.802 0.820

0.836

0.851 4 0.866 0.883 0.898 0.910

0.922

0.931

5

0.921 0.933 0.944 0.953

0.960

0.966 6 0.952 0.960 0.968 0.974

0.978

0.982 7 0.969 0.976 0.981 0.985

0.988

0.990 8 0.980 0.985 0.988 0.991

0.993

0.995

(44)

多時点のDunnett検定のαエラー

時点数

1

2

3

4

5

6

7

0.05

0.098 0.143 0.185 0.226

0.265 0.302

8

9

10

11

12

13

14 0.337 0.370

0.401 0.431 0.460 0.487 0.512

15

16

17

18

19

20

0.537 0.560 0.582 0.603 0.623

0.642

44

時点数5,10,20は2研究

(45)

例題

コレステロールデータ

対照群個体番号時点（週) 1 2 3 4 5 6 7 8 9 10 11 0 232 367 253 230 190 290 337 283 325 266 338 1 205 354 256 218 188 263 337 279 257 258 343 2 244 358 247 245 212 291 383 277 288 253 307 3 197 333 228 215 201 312 318 264 326 284 274 4 218 338 237 230 169 299 361 269 293 245 262 5 233 355 235 207 179 279 341 271 275 263 309 薬剤群個体番号時点（週) 21 22 23 24 25 26 27 28 29 30 31 32 0 317 186 377 229 276 272 219 260 284 365 298 274 1 260 166 375 208 290 230 190 225 236 284 281 225 2 255 170 348 242 286 230 216 244 221 274 276 242 3 250 115 314 202 289 235 219 248 222 267 280 243 4 254 177 318 194 280 208 222 227 223 291 260 215 5 246 185 314 205 244 230 201 234 221 282 270 226

(46)

46

(47)

(48)

48

p=0.075 p=0.021 p=0.058 p=0.047 p=0.005

（時点

5 群2）

(49)

2‐way経時型分散分析のモデル

データの構造：群，人，時点

要因：群，時点，群×時点

誤差：個体間誤差

個体内誤差

i：群

j：個体

ｋ：時点

ｙ

_ijk

＝μ＋α

_i

＋ε

_1iｊ

＋β

_k

＋αβ

_iｋ

＋ε

_2iｊk

μ：母平均

α

_i

_:群効果

β

_k

：時点効果

αβ

_iｋ

_{:群×時点交互作用}

(50)

50

2‐way経時型分散分析のモデル

対照群個体番号時点（週) 1 2 3 4 5 6 7 8 9 10 11 0 232 367 253 230 190 290 337 283 325 266 338 1 205 354 256 218 188 263 337 279 257 258 343 2 244 358 247 245 212 291 383 277 288 253 307 3 197 333 228 215 201 312 318 264 326 284 274 4 218 338 237 230 169 299 361 269 293 245 262 5 233 355 235 207 179 279 341 271 275 263 309 薬剤群個体番号時点（週) 21 22 23 24 25 26 27 28 29 30 31 32 0 317 186 377 229 276 272 219 260 284 365 298 274 1 260 166 375 208 290 230 190 225 236 284 281 225 2 255 170 348 242 286 230 216 244 221 274 276 242 3 250 115 314 202 289 235 219 248 222 267 280 243 4 254 177 318 194 280 208 222 227 223 291 260 215 5 246 185 314 205 244 230 201 234 221 282 270 226 群効果α

時点効果β

個体間誤差ε₁

α×

β

個体間誤差ε2

(51)

繰り返しのある

２元配置分散分析のモデル

B1 ｙ

₁₁₁

ｙ

₁₁₂

y

₁₁₃

・・・

A1 B2 ｙ

₁₂₁

ｙ

₁₂₂

y

₁₂₃

・・・

独立な繰り返し

B3 ｙ

₁₃₁

ｙ

₁₃₂

y

₁₃₃

・・・

要因：Ａ，Ｂ，Ａ×Ｂ

誤差：繰り返し誤差

i：Ａの水準

j：Ｂの水準

ｋ：繰り返し

ｙ

_ijk

＝μ＋α

_i

＋β

_j

＋αβ

_ij

＋ε

_ijk

誤差は１種類ε

₁

とε

₂

を一緒に評価

(52)

52

薬剤×時点の交互作用

薬剤 ns 時点 ns 薬剤×時点 ns 薬剤 * 時点 ns 薬剤×時点 ns 薬剤 ns 時点 * 薬剤×時点 ns 薬剤 * 時点 _* 薬剤×時点 _ns

(53)

53

薬剤×時点の交互作用

薬剤 _* 時点 _* 薬剤×時点 * 薬剤 *(ns) 時点 * 薬剤×時点 *

量的交互作用

質的交互作用

薬剤 * 時点 _* 薬剤×時点 _*

投与前値含む

(54)

54

0）個体差を無視した解析

proc mixed;class id time group;

model y=group time group*time;

Tests of Fixed Effects

Source

NDF DDF

TypeIIIF

Pr>F

GROUP

1

126

7.36 0.0076

TIME

5

126

1.13 0.3504

TIME*GROUP

5

126

0.25 0.9389

(55)

１）投与前値を含み、

ＣＳを仮定した解析

proc mixed;class id time group;

model y=group time group*time;

repeated /sub=id type=cs;run;

Tests of Fixed Effects

Source

NDF DDF

TypeIIIF

Pr>F

GROUP

1

21

1.36 0.2563

TIME

5

105

9.42 0.0001

(56)

56

各群の平均値

0

時点対照群薬剤群平均値の差

0

282.8

279.8 -3

1

268.9

247.5 -21.4

2

282.3

250.3 -32

3

268.4

240.3 -28.1

4

265.5

239.1 -26.4

5

267.9

238.2 -29.7

(57)

２）投与前値を除き、

ＣＳを仮定した解析

proc mixed;class id time group;

model y=group time group*time;

repeated/sub=id type=cs;where time ne 0; run;

Source

NDF DDF

TypeIIIF

Pr>F

GROUP

1

21

1.94 0.1784

TIME

4

84

2.59 0.0424

TIME*GROUP

4

84

0.31 0.8714

(58)

58

群平均と時点平均の推定

lsmeans group time/tdiff pdiff cl;

Effect TIME GROUP LSMEAN Std Error DF t Pr > |t| GROUP 1 270.6 14.3 21 18.96 0.0001 GROUP 2 243.1 13.7 21 17.79 0.0001 TIME 1 258.2 10.4 84 24.86 0.0001 TIME 2 266.3 10.4 84 25.64 0.0001 TIME 3 254.3 10.4 84 24.49 0.0001 TIME 4 252.3 10.4 84 24.29 0.0001 TIME 5 253.0 10.4 84 24.36 0.0001

Effect Difference Std Error DF t Pr > |t| Alpha Lower Upper

GROUP 27.5 19.8 21 1.39 0.1784 0.05 -13.6 68.6

(59)

３）投与前値との差に、

ＣＳを仮定した解析

proc mixed;class id time group;

model dif=group time group*time;

repeated/sub=id type=cs;where time ne 0; run;

Source

NDF DDF

TypeIIIF

Pr>F

GROUP

1

21

7.92 0.0104

TIME

4

84

2.59 0.0424

TIME*GROUP

4

84

0.31 0.8714

(60)

60 Result of 2‐way repeated measurement ANOVA(Post‐Pre) time： F=2.59，P=0.042 group： F=7.92，P=0.010 time× group： F=0.31，P=0.871 Means and SE bars are plotted. P‐value at each time point calculated by Student’s t test (no multiplicity adjustment) p=0.075 p=0.021 p=0.058 p=0.047 p=0.005

(61)

経時型分散分析のCommon Mistakes

1)薬剤効果を個体内変動を誤差として検定

薬剤効果：

個体を単位に割り付けるので

個体間変動を誤差にする

時点効果：

個体内で時点間比較を行うので

個体内変動を誤差にする

2)投与前値をそのままモデルに含めて解析

交互作用が有意となって解釈しにくい

３)投与前値が解析において考慮されてない．

４）分散構造の誤特定

(62)

62

経時データの共分散分析

投与前値Y

₁

の扱い

１）投与後の値(post：Y

₂

)と同様に単純に

モデルに含めて解析．

２）投与後の値だけで解析．

３）投与前値の差をとって解析．

４）投与前値との比をとって解析．

５）独立変数（共変量）としてモデルに含めて

解析．ANCOVA）．

i

Y

₂

=

μ

+

β

₁

⇒

₂

−

β

₁

=

μ

Y

₂

－ Y

₁

Y

₂

(63)

投与後値，差，共分散分析の比較

統計量

分散

post(Y

₂

)

σ

2

差（Y

₂

－ Y

₁

）

_σ

2

_+σ

2

－２

ρσ

2 ANCOVA

(Ｙ

₂

－βＹ

1

)

σ

2

－ρ

2

σ

2 母相関係数 ρ

ρ=0

ρ=0.5

ρ=１.0

post(Y

₂

）

_σ

2

σ

2

σ

2

差

（Y₂－ Y₁）

_2σ

2

_σ

₂

_０

ANCOVA

σ

2

_0.75σ

2

₀

[ ] [ ]

相関係数

の

2 1 2 2 1

,

:

Y

V

Y

V

ρ

σ

=

(64)

64

投与後差，共分散分析の比較

母相関係数ρ

分散

差

（Y₂－ Y₁）

post(Y

₂

）

ANCOVA

(65)

共分散分析の利点

ANCOVA(Ｙ

₂

－βＹ

₁

)

１）ρ＝0→ β=0

post(Y

₂

)と等価

２）ρ≒1→ β=1

差（Y

₂

－ Y

₁

）と等価

３）0<ρ<1

精度が高い

（Y

₂

－ Y

₁

）はρ>0.5以上ないと

post(Y

₂

)より精度が悪い

(66)

66

時点間のデータの相関構造

• 個体差が大きいため、時点間の測定値には

相関が生じる→通常はCSを仮定

• 複数の相関構造をあてはめ、結果に影響を

及ぼすか確認

• _GEEを適用

• 効果の推定、交互作用についての検討に重

点をおき、検定は参考とする．

(67)

分散構造

• CS(compound symmetry)

ある程度測定間隔が開いていて、直前の状

態が次に影響を与えず、個体間差が大きいと

きに用いる．

• _{empiricalオプション(GEE)}

仮定した分散構造に依存しないで、妥当な結

果が得られる．

(68)

68 t0 186 377 t1 166 375 t2 170 383 t3 115 333 t4 169 361 t5 179 355

ピンク：対照

赤：薬剤

(69)

時点間の相関行列

対照群 0 1 2 3 4 5 0 1.0000 0.9285 0.9025 0.8834 0.8916 0.9610 1 0.9285 1.0000 0.8964 0.7476 0.8185 0.9464 2 0.9025 0.8964 1.0000 0.8027 0.9512 0.9509 3 0.8834 0.7476 0.8027 1.0000 0.8874 0.8498 4 0.8916 0.8185 0.9512 0.8874 1.0000 0.9200 5 0.9610 0.9464 0.9509 0.8498 0.9200 1.0000 薬剤群 0 1 2 3 4 5 0 1.0000 0.8937 0.8305 0.8108 0.8793 0.9432 1 0.8937 1.0000 0.9610 0.8733 0.9379 0.9548 2 0.8305 0.9610 1.0000 0.9137 0.9056 0.9175 3 0.8108 0.8733 0.9137 1.0000 0.8694 0.8565 4 0.8793 0.9379 0.9056 0.8694 1.0000 0.9259 5 0.9432 0.9548 0.9175 0.8565 0.9259 1.0000

(70)

70

多次元空間の切断

(71)

CS(Compound

Symmetry)

複合対称性

６次元空間からどの２

次元空間を切断して

も同じ顔（相関係数）

になる．

どの時点でも

相関係数は等しい．

(72)

72

AR(Auto Regression)

自己相関構造

時点が離れると相関が小さくなる

1‐2の

相関

0.9 1‐3の

相関

0.8 1‐4の

相関

0.7 1‐5の

相関

0.6 1‐6の

相関

0.5

(73)

自己相関構造

y1 -11.6261 7.6004 y2 -12.5007 7.0590 y3 -10.4278 7.4829 y4 -11.5261 7.8749 y5 -9.2364 9.2239 y6 8.5839 1 .9 .8 .7 .6 .5 .9 1 .9 .8 .7 .6 .8 .9 1 .9 .8 .7 .7 .8 .9 1 .9 .8 .6 .7 .8 .9 1 .9 .5 .6 .7 .8 .9 1