ศᩓศᯒ⾲㻌 ANOVA Table

㸫5㸫 䠄 1 䠅䝰䝕䝹䠍 䠖㻌 ⾲䠑

3 ᅉᏊ䛾୰䚸2 䛴䛾ᅉᏊ䛿㧗䛔ᒙ䛻䛒䜛ሙྜ䛾䝰䝕䝹䠖 ゎᯒ䛷䛿䚸㧗䛔ᒙ䛾ᅉᏊ䜢᳨ᐃ䛩䜛䛯䜑䛾ㄗ ᕪ䠄୍ḟㄗᕪ䠅䛸ప䛔ᒙ䛾ᅉᏊ䜢᳨ᐃ䛩䜛ㄗᕪ䠄஧ḟㄗᕪ䠅䛾2✀㢮䛾ㄗᕪ䛜ᚲせ䛷䚸ศᩓศᯒ⾲䛿⾲5䛾 䜘䛖䛻䛺䜛䚹㧗䛔ᒙ䛾ᅉᏊ䠄୍ḟᅉᏊ䠅A䛸B䛻ᑐ䛧䛶䛿2ᅉᏊ஺஫స⏝AB䛷᳨ᐃ䛩䜛䚹䛣䛣䛷䛿㏙䜉䛺 䛔䛡䜜䛹䜒䚸䛣䛾䜘䛖䛺2䛴䛾ᒙ䛾䛒䜛ᐇ㦂䛷䛿䚸ᘧ1䜢⏝䛔䛶ィ⟬䛩䜛䛸䚸2ᅉᏊ஺஫స⏝AB䛷᳨ᐃ䛩 䜉䛝䛣䛸䛜ᑟ䛛䜜䜛䚹䜎䛯䚸ప䛔ᒙ䛻ᒓ䛩䜛ᅉᏊ䠄஧ḟᅉᏊ䠅䛿3ᅉᏊ஺஫స⏝䛷᳨ᐃ䛩䜛䚹⧞㏉䛧䛾䛺䛔 䝕䞊䝍䞉䝉䝑䝖䛷䛿䚸ᮏ᮶䛾஧ḟㄗᕪ䛿ᑟ䛡䛺䛔䛡䜜䛹䜒䚸3 ᅉᏊ䜘䜚㧗ḟ䛾஺஫స⏝䜢ㄗᕪ䛸ྠ⛬ᗘ䛾኱

䛝䛥䛸䜏䛺䛫䜛䛣䛸䛜䚸⤒㦂ୖ᫂䜙䛛䛻䛺䛳䛶䛔䜛䛛䜙䛷䛒䜛䚹

䠄 2 䠅䝰䝕䝹2 䠖㻌 ⾲䠒

3ᅉᏊ䛸䜒䚸ྠ୍䛾ᒙ䛻䛒䜛ሙྜ䠄ὀཧ↷䠅䛾䝰䝕䝹䠖ゎᯒ䛷䛿䚸ㄗᕪ䛜1䛴䛾ศᩓศᯒ⾲䠄⾲6䠅䛻䛺䜛䚹 ኚືᅉ ⮬⏤ᗘ ೫ᕪᖹ᪉࿴ ᖹᆒᖹ᪉࿴ F₀್ P್䠄☜⋡䠅 ุᐃ

඲య

df

_x

S

_䡔

S

_䡔/

df

_x

ᅉᏊ㻌A

df

A

S

A

a = S

A

/ df

A

a / 䡀

^շ

ᅉᏊ㻌B

df

_B

S

_B

b = S

_B

/ df

_B

b / 䡀

ո

஺஫స⏝AB㻌 㻌 䠄୍ḟㄗᕪ䠅

df

_AB

S

_AB

d = S

_AB

/ df

_AB

d /g

մ

ᅉᏊ㻌C

df

_C

S

_C

c = S

_C

/ df

_C

c /g

ճ

஺஫స⏝AC

df

AC

S

AC

e = S

AC

/ df

AC

e /g

յ

஺஫స⏝BC

df

BC

S

BC

f = S

BC

/ df

BC

f /g

ն

஺஫స⏝ABC 䠄஧ḟㄗᕪ䠅

df

_ABC

S

_ABC

= S

_E

g = S

_ABC

/ df

_ABC

ኚືᅉ ⮬⏤ᗘ ೫ᕪᖹ᪉࿴ ᖹᆒᖹ᪉࿴ F₀್ P್䠄☜⋡䠅 ุᐃ

඲య

df

_x

S

_䡔

S

_䡔/

df

_x

ᅉᏊ㻌A

df

A

S

A

a = S

A

/ df

A

a /g

ձ

ᅉᏊ㻌B

df

B

S

B

b = S

B

/ df

B

b /g

ղ

ᅉᏊ㻌C

df

_C

S

_C

c = S

_C

/ df

_C

c /g

ճ

஺஫స⏝AB

df

AB

S

AB

d = S

AB

/ df

AB

d /g

մ

஺஫స⏝AC

df

_AC

S

_AC

e = S

_AC

/ df

_AC

e /g

յ

஺஫స⏝BC

df

BC

S

BC

f = S

BC

/ df

BC

f /g

ն

஺஫స⏝ 㻭㻮㻯㻌 䠄ㄗᕪ䠅

df

_ABC

S

_ABC

= S

_E

g = S

_ABC

/ df

_ABC

⾲5 3ඖ㓄⨨ศᩓศᯒ⾲㸦3ᅉᏊࡢෆ2ࡘࡢᅉᏊA䛸Bࡣ㧗࠸ᒙࠊCࡣప࠸ᒙࡢࣔࢹࣝ㸧ࣔࢹࣝ㸯

⾲㸴 3ඖ㓄⨨ศᩓศᯒ㸦3ᅉᏊ࡜ࡶྠ୍ᒙࡢࣔࢹࣝ㸧 ࣔࢹࣝ2

䠄3䠅䝰䝕䝹3 䠖㻌 ⾲䠓

3ᅉᏊ䛾୰䚸1ᅉᏊ䛰䛡䛿㧗䛔ᒙ䛻䛒䜛䠄୍ḟᅉᏊ䛸࿧䜆䠅ሙྜ(ὀ) 䛾䝰䝕䝹䠖ゎᯒ䛷䛿䚸㧗䛔ᒙ䛾ᅉᏊ 䜢᳨ᐃ䛩䜛䛯䜑䛾ㄗᕪ䠄୍ḟㄗᕪ䠅䛸ప䛔ᒙ䛾ᅉᏊ䠄஧ḟᅉᏊ䠅䜢᳨ᐃ䛩䜛ㄗᕪ䠄஧ḟㄗᕪ䠅䛾2✀㢮䛾ㄗᕪ 䛜ᚲせ䛷䛒䜛䛜䚸௨ୗ䛾ศᩓศᯒ䛷䛿୍ḟㄗᕪ䛜ᚓ䜙䜜䛺䛔䛾䛷䚸䛩䜉䛶䛾ᅉᏊ䜢஧ḟㄗᕪ䛷᳨ᐃ䛩䜛䚹 䛯䛰䛧䚸୍⯡䛻୍ḟㄗᕪศᩓ䠚஧ḟㄗᕪศᩓ䛾㛵ಀ䛜䛒䜛䛛䜙䚸䛣䛾᳨ᐃ䛷㧗䛔఩⨨䛾ᅉᏊ䛜䜒䛧᭷ព䛸

ุᐃ䛥䜜䛶䜒䚸ᮏᙜ䛻᭷ព䛰䛳䛯䛛䛹䛖䛛䛿ศ䛛䜙䛺䛔䛸䛔䛖ḞⅬ䛜䛒䜛䚹ศᩓศᯒ⾲䛿⾲7䛾䜘䛖䛻䛺䜛䚹

䠄ὀ䠅㻌 䛂㧗䛔ᒙ䛃䛸䛂ప䛔ᒙ䛃 䠖 ⡆༢䛾䛯䜑䛻䚸2 ᅉᏊ䛰䛡䛾ᐇ㦂䜢⪃䛘䜘䛖䚹䛔䛟䛴䛛䛾௻ᴗ㛫䛷䛒䜛✀䛾

୙Ⰻရ䛾๭ྜ䜢ẚ㍑䛩䜛ሙྜ䚸௻ᴗෆ䛻䛚䛔䛶⏨ዪ(ᛶᕪ)䜒ᅉᏊ䛸䛧䛶ྲྀ䜚ୖ䛢䛯䛸䛩䜛䚹䛣䛾ሙྜ䚸௻ᴗ 䛿䛭䛾༢఩䛸䛧䛶ᛶᕪ䜢ෆ䛻ྵ䜣䛷䛔䜛䚹ᅉ䛳䛶䚸௻ᴗ䛿㧗䛔ᒙ䛾ᅉᏊ䠄୍ḟᅉᏊ䛸࿧䜀䜜䜛䠅䚸ᛶᕪ䛿ప 䛔ᒙ䛾ᅉᏊ䠄஧ḟᅉᏊ䠅䛸ゝ䛘䜛䚹

⾲5䛚䜘䜃⾲䠒䛾P್㻌 䠄㻌 ⤫ィゎᯒྥ䛡ゝㄒ㻌R䛾䡌䡂 㛵ᩘ䛾ᘬᩘ䜢♧䛩㻌 䠅㻌

No. P ್䠙pf 䠄F0್䠈⮬⏤ᗘ䠍䠈⮬⏤ᗘ 䠎䠅 No. P ್䠙pf 䠄F0್䠈⮬⏤ᗘ䠍䠈⮬⏤ᗘ䠎䠅 ձ

pf ( a/g, 䡀䡂

A

, df

_ABC

)

յ

pf ( e/g, 䡀䡂

AC

, df

_ABC

)

ղ

pf ( b/g, 䡀䡂

B

, df

ABC

)

ն

pf ( f/g, 䡀䡂

BC

, df

ABC

)

ճ

pf ( c/g, 䡀䡂

C

, df

_ABC

)

շ

pf ( a/d, 䡀䡂

A

, df

_AB

)

մ

pf ( d/g, 䡀䡂

AB

, df

ABC

)

ո

pf ( b/d, 䡀䡂

B

, df

AB

)

5䠊䝰䝕䝹䝕䞊䝍䜢⏝䛔䛯䛂R䛻䜘䜛3ඖ㓄⨨ศᩓศᯒ䝅䝇䝔䝮䛃䛾᳨ド

3ಶ䛾ᅉᏊ䛛䜙ᡂ䜛㻌3ඖ㓄⨨ᵓ㐀䛾ゎᯒ౛䠄⾲9䡚10䠅䜢ᮏ◊✲䛷㛤Ⓨ䛧䛯䛂R䛻䜘䜛3ඖ㓄⨨ศᩓศ ᯒ䝅䝇䝔䝮䛃䛷ศᩓศᯒ䜢⾜䛳䛯䠄 ⾲11䡚12 䠅䚹

ኚືᅉ ⮬⏤ᗘ ೫ᕪᖹ᪉࿴ ᖹᆒᖹ᪉࿴ F₀್ P್䠄☜⋡䠅 ุᐃ

඲య dfx

S

_䡔

S

_䡔/

df

x

ᅉᏊ㻌 A

df

_A

S

_A

a = S

_A

/ df

_A

a /g

ձ

ᅉᏊ㻌 B

df

_B

S

_B

b = S

_B

/ df

_B

b /g

ղ

஺஫స⏝AB㻌

df

AB

= 0 S

AB

= 0 d = S

AB

/ df

AB

䠄୍ḟㄗᕪ䠅

ᅉᏊ㻌 C

df

C

S

C

c = S

C

/ df

C

c /g

ճ

஺஫స⏝AC

df

_AC

S

_AC

e = S

_AC

/ df

_AC

e /g

յ

஺஫స⏝BC

df

_BC

S

_BC

f = S

_BC

/ df

_BC

f /g

ն

஺஫స⏝ABC 䠄஧ḟㄗᕪ䠅

df

ABC

S

ABC

g = S

ABC

/ df

ABC

⾲8 P^{್ࡢ⟬ฟἲ}

⾲㸵 3ඖ㓄⨨ศᩓศᯒ⾲㸦3ᅉᏊࡢෆࠊ2ᅉᏊࡀ㧗࠸ᒙࡢࣔࢹࣝ㸧ࣔࢹࣝ3

ᣦᐃࡍࡿ୍ḟㄗᕪࡀ࡞࠸ࡢ࡛ࠊ஧ḟㄗᕪ᳨࡛ᐃࡍࡿࡀࠊF0್ࡣ㐣኱ホ౯࡜࡞ࡾࠊ᭷ព࡛࡞࠸ࡢ࡟᭷ព࡜ุᐃ ࡉࢀࡿ༴㝤ᛶࡀ࠶ࡿࠋ

㸫7㸫 䠄䠍䠅㻌3ඖ㓄⨨ศᩓศᯒ䛾౛

㻌 䠄a) ゎᯒ౛䠄䠍䠅

⾲9࡟㻌 ᅉᏊA䠄5Ỉ‽䠅䚸㻌 ᅉᏊB䠄6Ỉ‽䠅䚸㻌 ᅉᏊC䠄2Ỉ‽䠅㻌 䛾ヨ㦂䝕䞊䝍(1)䜢♧䛩䚹䛩䛺䜟䛱䚸 䛒䜛〇ရ䜢6♫䛻䛚䛔䛶〇㐀䛧䛶䛔䜛䚹⾲9䛿2006ᖺ䛛䜙2010ᖺ䜎䛷䛾5ᖺ㛫䛾〇ရ䛾ᢤ䛝ྲྀ䜚᳨ᰝ䛾

⤖ᯝ䚸䛒䜛つ᱁௨ୗ䛾〇ရ䛾๭ྜ䠄䠂䠅䜢♧䛧䛶䛔䜛䚹᳨ᰝဨ䛻䛿⏨ᛶ䛸ዪᛶ䛜䛔䛶䚸ᛶ䛻䜘䛳䛶䜒ᕪ䛜䛒 䜛䛛䛹䛖䛛䜒᫂䜙䛛䛻䛧䛯䛔䚹䛣䛾ሙྜ䚸䛂ᖺḟ䛃䛸䛂௻ᴗ䛃䛿䛸䜒䛻䛂ᛶ䛃䜢༢఩䛸䛧䛶ྵ䜐䛾䛷䛂ᛶ䛃䜘䜚㧗䛔ᒙ 䛻䛒䜛䛸⪃䛘䜛䚹

㻌 䠄b) ゎᯒ౛䠄2䠅

⾲10࡟㻌 ྠ୍ᒙ䛾3䛴䛾ᅉᏊ䛾ᅉᏊA 䠄3Ỉ‽䠅䚸㻌 ᅉᏊB 䠄3Ỉ‽䠅䚸㻌 ᅉᏊC 䠄3Ỉ‽䠅㻌 䛾ヨ㦂䝕䞊䝍䠄2䠅䜢♧䛩䚹⫧ᩱ䛾୕せ⣲䛷䛒䜛

❅⣲⫧ᩱ䚸䝸䞁㓟⫧ᩱ䚸䜹䝸⫧ᩱ䜢3ᅉᏊ䛸䛧䛶 䝃䝖䜴䜻䝡䛾ྍ〇⢾⋡ࢆồࡵࡓゎᯒ౛࡛࠶ࡿࠋ

⫧ᩱࡢ⤌ࡳྜࢃࡏ27ಶࡢヨ㦂༊ࡣࠊࣛࣥࢲ࣒

࡞㡰ᗎ࡛⏿࡟᱂ᇵࡉࢀࡓࠋ

㸦㸰㸧R࡟ࡼࡿ3ඖ㓄⨨ศᩓศᯒ䛾⤖ᯝ 䠄䠽䠅䝰䝕䝹1䠄 ⾲5 䠅䜢ᇶᗏ䛻䛧䛶⾲䠕䛾

䝕䞊䝍䜢ศᩓศᯒ䛧䛯⤖ᯝ䠄⾲11䠅

3䛴䛾ᅉᏊ䛾୰䚸䛂ᖺ䛃䛸䛂௻ᴗ䛃䛾2䛴䛾ᅉᏊ

䛿㧗䛔ᒙ䛻䛒䜚䚸ᅉᏊ䛂ᛶู䛃䛿ప䛔ᒙ䛻䛒䜛䚹 㧗䛔ᒙ䛾ᅉᏊ䜢᳨ᐃ䛩䜛䛯䜑䛾ㄗᕪ䠄୍ḟㄗ ᕪ䠅䛸ప䛔ᒙ䛾ᅉᏊ䜢᳨ᐃ䛩䜛ㄗᕪ䠄஧ḟㄗᕪ䠅 䛾2✀㢮䛾ㄗᕪ䛜Ꮡᅾ䛩䜛䚹

ᅉᏊC䛚䜘䜃஺஫స⏝AB䛿ᖐ↓௬ㄝ䛻ᑐ 䛧䛶䚸5䠂䛾༴㝤⋡䛷᭷ព䛸ุᐃ䛥䜜䛯䚹䜎䛯䚸

ᅉᏊB䛿5䠂䛾༴㝤⋡䛷䚸᭷ព䛻㏆䛔䛸ุᐃ䛥

䜜䛯䚹

⾲㸷 3ඖ㓄⨨ᵓ㐀ࡢヨ㦂ࢹ࣮ࢱ(1)

⾲10 3ඖ㓄⨨ᵓ㐀ࡢヨ㦂ࢹ࣮ࢱ(2)

㻌ns : not significant䠄᭷ព䛷䛺䛔䠅 䠄b䠅㻌 䝰䝕䝹2䠄⾲䠒䠅䜢ᇶᗏ䛻䛧䛶⾲10䛾䝕䞊䝍䜢ศᩓศᯒ䛧䛯⤖ᯝ䠄⾲12䠅

3ᅉᏊ䛸䜒䚸஫䛔䛻ྠ୍䛾ᒙ䛻䛒䜛䛾䛷ㄗᕪ䛜䠍䛴䛾ศᩓศᯒ⾲䠄⾲12䠅䛻䛺䜛䚹ᅉᏊA䠄❅⣲⫧ᩱ䠅䛾 䜏䛜ᖐ↓௬ㄝ䛻ᑐ䛧䛶䚸᭷ព䛸ุᐃ䛥䜜䛯䚹ᅉᏊB䠄䝸䞁㓟⫧ᩱ䠅䚸ᅉᏊC䠄䜹䝸⫧ᩱ䠅䛚䜘䜃3䛴䛾஺஫స

⏝䛿ᖐ↓௬ㄝ䛻ᑐ䛧䛶䚸᭷ព䛷䛿䛺䛔䛸ุᐃ䛥䜜䛯䚹

ns : not significant䠄᭷ព䛷䛺䛔䠅

6䠊㻌 ᭱ᑠ᭷ពᕪ㻌LSD (Least Significant Difference )

㻌 ⾲5 䡚 7䛾ศᩓศᯒ⾲䛾P ್䛜᭷ព䛸ุᐃ䛥䜜䛯ሙྜ䛻䛿䚸ᙜヱᅉᏊෆ䛾Ỉ‽㛫䛻᭷ពᕪ䛜䛒䜛䛣䛸 䜢♧䛧䛶䛔䜛䚹䛧䛛䛧䚸䛹䛾Ỉ‽㛫䛻᭷ពᕪ䛜䛒䜛䛛䛹䛖䛛䛿ศᩓศᯒ䛷䛿ุᐃ䛷䛝䛺䛔䚹䛣䛾᭱ᑠ᭷ពᕪ LSD䛾⟬ฟ䛻䛿䚸䝔䝳䞊䜻䞊䛾᪉ἲ䛾ᩘ⾲䠄䝇䝏䝳䞊䝕䞁䝖໬䛧䛯⠊ᅖ䛾䝟䞊䝉䞁䝖Ⅼ䠅q䜎䛯䛿R䛾㛵ᩘ

䠄qtukey䠅䜢⏝䛔䜛䚹䛣䛣䛷䚸⾲11䛾ศᩓศᯒ䛾⤖ᯝ䜢౑䛳䛶LSD䜢᳨ウ䛩䜛䚹 䠄䠍䠅ᅉᏊ䛾᭱ᑠ᭷ពᕪLSD㻌 䠖㻌 ᅉᏊB䛾౛䛷ㄝ᫂䛩䜛

LSD䠙q䠄Ỉ‽䛾ᩘ䠄6䠅䚸ㄗᕪ䛾⮬⏤ᗘ䠄20䠅䚸༴㝤⋡䠄0.05䠅䠅䠆䠄ᅉᏊB䛾ᶆ‽ㄗᕪ䠅/ 2

ᅉᏊB䛾ᶆ‽ㄗᕪ 䠙 ㄗᕪࡢᖹᆒᖹ᪉࿴(2.763) /(AࡢỈ‽ᩘ(5))(CࡢỈ‽ᩘ(2)) 䠙 0.5257 䠄䛣䛣䛷䚸ㄗᕪ䛾ᖹᆒᖹ᪉࿴䛸䛧䛶䚸୍ḟㄗᕪ䜢⏝䛔䛯䛣䛸䛻ὀព䠅 䛣䛣䛻䚸q䠄ᅉᏊB䛾Ỉ‽䛾ᩘ(6)䚸ㄗᕪ䛾⮬⏤ᗘ(20)䚸༴㝤⋡(0.05䠅=4.452

ᚑ䛳䛶䚸LSD 䠙 4.452 * 0.5257/ (2^0.5) = 1.652䚹㻌 ᨾ䛻䚸⾲9䛾ᅉᏊB䛾6䛴䛾Ỉ‽䛾ᖹᆒ㛫䛾ᕪ䛜1.652 ௨ୖ䛾䜒䛾䛿Ỉ‽㛫䛷᭷ព䛷䛒䜛䚹༶䛱䚸⾲13䛻䜘䜜䜀Ỉ‽䞉௻ᴗ䠏䛿௻ᴗ䠐௨እ䛾䛩䜉䛶䛾Ỉ‽䛸䛾ᖹ ᆒ䛾ᕪ䛜LSD䜘䜚䜒኱䛝䛔䚹

⾲12 R࡟ࡼࡿ3ඖ㓄⨨ᵓ㐀ࡢࢹ࣮ࢱࡢศᩓศᯒ⤖ᯝ㸦ㄗᕪࡀ1ࡘࡢሙྜ㸧

⾲11 R࡟ࡼࡿ3ඖ㓄⨨ᵓ㐀ࡢࢹ࣮ࢱࡢศᩓศᯒ⤖ᯝ㸦ㄗᕪࡀ୍ḟ࡜஧ḟࡢ2ࡘࡢሙྜ㸧

ศᩓศᯒ䛷᭷ព䛺ᅉᏊC䛿2Ỉ‽䛷䛒䜛䛛䜙Ỉ‽㛫䛷᭷ពᕪ䛜䛒䜛䛣䛸䛿᫂☜䛷䛒䜛䛾䛷ᖹᆒ䛾䠄ከ㔜䠅 ẚ㍑䜢⾜䛖ᚲせ䛿䛺䛔䛜䚸䜒䛧䚸䛣䛣䛷䜒LSD䜢ィ⟬䛩䜜䜀䚸LSD = 0.247䛸䛺䜛䚹䛣䛾᫬䛾ㄗᕪ䛿2ḟㄗᕪ 䜢⏝䛔䜛䚹[ゎᯒ౛(1)㸸⾲9 ] ࠿ࡽᑟฟࡋࡓᅉᏊCࡢỈ‽㸦⏨ᛶ㸧ࡢᖹᆒࡣ2.344ࠊỈ‽㸦ዪᛶ㸧

ࡢᖹᆒࡣ3.080࡛ࡑࡢᕪࡣ0.736࡛࠶ࡿ࠿ࡽࠊLSDࡼࡾ኱ࡁࡃ᭷ព࡜ุᐃ࡛ࡁࡿࠋ

䠄䠎䠅஺஫స⏝ AB䛾᭱ᑠ᭷ពᕪLSD

2ᅉᏊ஺஫స⏝䛻ᑐ䛩䜛LSD䛿䚸⾲13䛸14䛷䛿ᖹᆒ䜢ᵓᡂ䛧䛶䛔䜛䝕䞊䝍ᩘ䛜␗䛺䛳䛶䛔䜛䛾䛷䚸୺

ຠᯝ䛻ᑐ䛩䜛LSD䛸䛿␗䛺䜛䛣䛸䛻ὀព䜢せ䛩䜛䚹2ᅉᏊ஺஫స⏝AB䛻ᑐ䛩䜛LSD䛿ḟ䛾㏻䜚䛷䛒䜛䚹 㻌 㻌LSD䠙q䠄䠄ᅉᏊA䛾Ỉ‽ᩘ(5)䠅䠆䠄ᅉᏊBỈ‽ᩘ(6)䠅䚸ㄗᕪ䛾⮬⏤ᗘ䠄20䠅䚸༴㝤⋡䠄0.05䠅䠅

䠆䠄ㄗᕪ䛾ᶆ‽ㄗᕪ䠅/ 2

ㄗᕪ䛾ᶆ‽ㄗᕪ䠙 ୍ḟㄗᕪࡢᖹᆒᖹ᪉࿴(2.7632)/ᅉᏊCࡢỈ‽ᩘ(2) 1.1754 ᚑ䛳䛶䚸LSD 䠙 6.1039 * 1.1754/(2^0.5) = 5.0739䚹⾲14䛻䜘䜜䜀௻ᴗ䠍䛿2009ᖺḟ䛸2010ᖺḟ䛾䝕䞊䝍 䛾ᕪ䛜LSD䜘䜚኱䛝䛟᭷ព䛷䛒䜛䚹䜎䛯䚸௻ᴗ2䛿2007ᖺḟ䛸2010ᖺḟ䛸䛾䝕䞊䝍䛾ᕪ䛜LSD䜘䜚኱䛝䛟

᭷ព䛷䛒䜛䚹

7䠊⪃ᐹ

⾲1䛻♧䛩3ඖ㓄⨨ศᩓศᯒ䛾䝕䞊䝍䛾ᆺ䛻ᑐ䛩䜛ศᩓศᯒ䛾䝥䝻䜾䝷䝮䜢㛤Ⓨ䛧䛯䚹䛣䛾䜘䛖䛺3ඖ 㓄⨨䛾ᆺ䛾䝕䞊䝍䜢ศᩓศᯒ䝥䝻䜾䝷䝮䛿Excel䛾䜰䝗䜲䞁䝋䝣䝖䠄ศᯒ䝒䞊䝹䠅䛻䛿ྵ䜎䜜䛶䛔䛺䛔䚹

ᐇ౑⏝䛻⪏䛘䜛䜉䛟䚸⾲5䡚7䛾3✀䛾ศᩓศᯒ⾲䜢⟬ฟྍ⬟䛻䛧䛯䚹⾲9䡚10䛻♧䛩㻌3ඖ㓄⨨ᵓ㐀 䛾ゎᯒ౛䜢ศᩓศᯒ䛧䛯⤖ᯝ䜢⾲11䡚12䛻♧䛧䛯䚹⾲11䛷䛿䚸2ᅉᏊ஺஫స⏝AB䜢୍ḟㄗᕪ䛸䛧䚸3ᅉ Ꮚ஺஫స⏝ABC䜢஧ḟㄗᕪ䛸䛧䛯䚹䛭䛾⤖ᯝ䚸ᅉᏊB䛸C䛚䜘䜃AB䛜᭷ព䛾ุᐃ䛻䛺䛳䛯䚹⾲12䛷䛿 3ᅉᏊ஺஫స⏝ABC䛾䜏䜢ㄗᕪ䛸䛧䛶ᢅ䛳䛯䚹䛭䛾⤖ᯝ䚸⾲12䛻♧䛩ዴ䛟䚸ᅉᏊA䛾䜏䛜᭷ព䛾ุᐃ䛻 䛺䛳䛯䚹

䝥䝻䜾䝷䝮䛾☜ㄆ䛾䛯䜑䛻䚸⾲9䡚10䛾ゎᯒ౛䛻ຍ䛘䛶䚸⚟⏣䞉༲᭶䠄2009) 䛂ᨺ㏦኱Ꮫ኱Ꮫ㝔༳ๅᩍᮦ

䠄ᚰ⌮䞉ᩍ⫱䛾⤫ィ≉ㄽ㻌pp57-69䠅䛃䛚䜘䜃䚸R.MEAD(1992) 䛂The㻌design of experiments䠄pp. 359-361䠅䛃䛾 䝕䞊䝍䜢ศᩓศᯒ䛧䛯⤖ᯝ䚸ᩥ⊩䛻♧䛥䜜䛯ゎᯒ⤖ᯝ䛸୍⮴䛧䛯䚹㻌

⾲13 [ゎᯒ౛(1)㸸⾲9 ] ࠿ࡽᑟฟࡋࡓᅉᏊ㹀ࡢỈ‽ࡢᖹᆒ

⾲14 [ゎᯒ౛(1)㸸⾲9 ] ࠿ࡽᑟฟࡋࡓᅉᏊA࡜ᅉᏊ㹀ࡢ2ඖ⾲ 㸦⾲2࡟ᑐᛂࡍࡿ㸧

䜎䛯䚸⾲11 䛻♧䛩ศᩓศᯒ䛾⤖ᯝ䚸᭷ព䛸ุᐃ䛥䜜䛯ᅉᏊ䛾Ỉ‽㛫䛾᭱ᑠ᭷ពᕪ㻌LSD 䜢ィ⟬䛧䚸ල యⓗ䛻᭷ព䛷䛒䜛Ỉ‽㛫䜢ᣦ᦬䛧䛯䚹

8䠊䛚䜟䜚䛻 䠄䠍䠅䜎䛸䜑

㻌 㻌3ඖ㓄⨨ศᩓศᯒ䛿Excel䜔Rゝㄒ䛾㛵ᩘ䛿ഛ䛘䜙䜜䛶䛔䛺䛔䚹ᮏ◊✲䛷䛿3ඖ㓄⨨ศᩓศᯒ䜢R ゝㄒ䛷䝥䝻䜾䝷䝭䞁䜾䛧䛯䚹䜎䛯䚸3ඖ㓄⨨䛾✀䚻䛾䝕䞊䝍䜢௒ᅇ㛤Ⓨ䛧䛯䛂R䛻䜘䜛3ඖ㓄⨨ศᩓศᯒ䝅䝇 䝔䝮䛃䛷ศᩓศᯒ䛧䚸3ඖ㓄⨨䛾䝕䞊䝍䛻㐺⏝䛷䛝䜛䝅䝇䝔䝮䛷䛒䜛䛣䛸䜢☜ㄆ䛧䛯䚹

䠄䠎䠅ᮏ◊✲䛾㛵㐃㈨ᩱ䛾Web䝃䞊䝞䜈䛾᱁⣡

ᮏ◊✲䛷㛤Ⓨ䛧䛯Rゝㄒ䛾䝇䜽䝸䝥䝖䠄䝥䝻䜾䝷䝮䠅䛿ᐜ㔞䛜኱䛝䛔䛾䛷䚸䛣䛣䛻グ䛩䛣䛸䛿䛷䛝䛺䛔䚹ᚑ䛳 䛶䛂R䛻䜘䜛3ඖ㓄⨨ศᩓศᯒ䝅䝇䝔䝮䛾䝇䜽䝸䝥䝖䠄䝥䝻䜾䝷䝮䠅䛃䜢ୗグ䛻Ⓩ㘓䛧䛯䚹

https://sites.google.com/a/yokappe.net/mathematics310/

ⲈᇛSC䝣䜷䞊䝷䝮

௒ᚋ䛾ㄢ㢟

ୗグ䝰䝕䝹䜢ྵ䜐䜘䜚୍⯡ᆺ䛾3ඖ㓄⨨ศᩓศᯒ⏝䝥䝻䜾䝷䝮䜢㛤Ⓨ䛩䜛䚹 䠄䠍䠅㻌3ᅉᏊ䛩䜉䛶䛜3䛴䛾ู䚻䛾ᒙ䛻ᒓ䛧䛶䛔䜛ሙྜ䛾䝰䝕䝹

䠄䠎䠅㻌1ᅉᏊ䛿㧗䛔ᒙ䛻ᒓ䛧䚸௚䛾2ᅉᏊ䛿ప䛔ᒙ䛻ᒓ䛩䜛ሙྜ䛾䝰䝕䝹 ㅰព

ᮏ◊✲䛻ᙜ䛯䜚䚸ⲈᇛᏛ⩦䝉䞁䝍䞊䛾⤫ィᏛ䜾䝹䞊䝥䛾䜏䛺䛥䜣䛻䚸ከ䛟䛾ሙ䛷㆟ㄽ䛻ཧຍ䛔䛯䛰䛝䚸 䛤ᣦᑟ䜢㈷䛳䛯䚹

ཧ⪃ᩥ⊩

㸦㸯㸧⚟⏣࿘࣭༲᭶◊ḟ㸪2009㸪ᚰ⌮࣭ᩍ⫱ࡢ⤫ィ≉ㄽ㸪ᨺ㏦኱Ꮫᩍ⫱᣺⯆఍㸪ᮾி㸪pp㸬57-69 㸦㸰㸧 R. MEAD㸪1992㸪The design of experiments: statistical principles for practical applications, Cambridge

University Press㸪New York㸪pp . 357-361

䊢䠉䠏

㻌

ᑠᏛᰯ㛗ᮇ⮬↛య㦂άື䛻䛚䛡䜛⏕䛝䜛ຊ䛾⪃ᐹ㻌 䇷㻌 䠍㐌㛫䛾᫬⣔ิཬ䜃ᛶᕪ䛾ほⅬ䛛䜙䛾ศᯒ㻌 䇷㻌

ᨺ㏦኱Ꮫ኱Ꮫ㝔㻌 ಟኈ㑅⛉⏕㻌 ⱱ࿅㍤ኵ㻌

䠍䠊 ┠ⓗ㻌

㻌 ᖹᡂ䠎䠌ᖺ䠄㻞㻜㻜㻤䠅䠍᭶䠍䠓᪥䚸୰ኸᩍ⫱ᑂ㆟఍䛾⟅⏦䛂ᗂ⛶ᅬ䚸ᑠᏛ䚸୰Ꮫᰯ䚸㧗➼Ꮫᰯཬ䜃≉ูᨭ᥼

Ꮫᰯ䛾Ꮫ⩦ᣦᑟせ㡿➼䛾ᨵၿ䛻䛴䛔䛶䛃^䠍䠅㻌 䛜Ⓨ⾲䛥䜜䚸Ꮫ⩦ᣦᑟせ㡿䛾ᨵゞ䛾䛽䜙䛔䛸䛭䛾ᴫせ䛜᫂

☜䛻䛥䜜䛯䚹䛭䛣䛷䚸Ꮚ䛹䜒䛯䛱䛾䛂⏕䛝䜛ຊ䛃䜢⫱䜐䛣䛸䛾ᚲせᛶ䜔䛂⏕䛝䜛ຊ䛃䛾ෆᐜ䜢ᩍ⫱㛵ಀ⪅䜔ಖ ㆤ⪅䚸♫఍䛜⮬䜙⪃䛘䚸⌮ゎ䛾ୖඹ᭷䛩䜛䛣䛸䛿䚸௒ᅇ䛾Ꮫ⩦ᣦᑟせ㡿ᨵゞ䛻㝿䛧䛶䜎䛪⾜䜟䛺䛡䜜䜀 䛺䜙䛺䛔䛸䛥䜜䛯䚹㻌

䛣䛾䜘䛖䛺୰䛷䚸㔝እάື䜢୰ᚰ䛸䛧䛯஦ᴗሗ࿌䛜ྛ⮬἞య䜔ᅜ䛾ᩍ⫱ᶵ㛵䛛䜙䛺䛥䜜䛶䛔䜛䛜䚸ྛ

஦ᴗ䛾ᢸᙜ⪅䛜సᡂ䛧䛯⡆༢䛺䜰䞁䜿䞊䝖ㄪᰝ䛜ᐇ᪋䛥䜜䛶䛔䜛⛬ᗘ䛷䚸ホ౯ᇶ‽䜒ྛᐇ᪋ᅋయ䜔஦

ᴗ䛤䛸䛻㐪䛳䛶䛔䜛䛾䛜ᐇែ䛷䛒䜛䚹䛣䜜䛷䛿䚸ⓗ☜䛻஦ᴗ䛾ᡂᯝ䜢䛸䜙䛘䛶䛔䜛䛸䛿ゝ䛔䛜䛯䛔䚹㻌 㔝እᩍ⫱㛵ಀ䛷䛿䚸ෆⓎⓗືᶵ௜䛡䜔⮬ᕫຠຊឤ䛺䛹䛾ᚰ⌮Ꮫⓗᑻᗘ䜢⏝䛔䛯◊✲䛜ሗ࿌䛥䜜䛶䛔

䜛䚹^{䠎䠅䠏䠅䠐䠅}㻌 䛣䜜䜙䛾◊✲䛻䜘䜚䚸㔝እάື䜔㛗ᮇ⮬↛య㦂䛾᭷ຠᛶ䜔ᩍ⫱ⓗព⩏䛻䛴䛔䛶䛿☜ㄆ䛥䜜䛶䛝

䛯䚹^䠑䠅㻌 䛭䛾䜘䛖䛺୰䛷䚸ᶲ䜙䛿䛂⏕䛝䜛ຊ䛃䛻╔┠䛧䚸㡯┠䛾཰㞟䛸⢭㑅䜢⾜䛔䚸ศᯒ䜢⾜䛳䛯䚹䛭䛾⤖ᯝ䚸 䛂⏕䛝䜛ຊ䛃䛿ᚰ⌮ⓗ♫఍ⓗ⬟ຊ䞉ᚨ⫱ⓗ⬟ຊ䞉㌟యⓗ⬟ຊ䛾䠏䛴䛾ୖ఩ᑻᗘ䛸䠍䠐䛾ୗ఩ᑻᗘ䛛䜙ᡂ䜛䛸 䛧䛶䠥䠧䠮ホᐃ⏝⣬䜢సᡂ䛧䛯䚹^䠒䠅㻌 䛭䛾ᑻᗘ䜢⏝䛔䛶䚸ᶲ䜙䛿ᐇ㝿䛾㛗ᮇ䜻䝱䞁䝥䛻ཧຍ䛧䛯ᑠ୰Ꮫ⏕䛾 ᖹᆒᚓⅬ䜢ึ᪥䛸᭱⤊᪥䛷ẚ㍑䛧䛯䛸䛣䜝䚸䛩䜉䛶䛾㡯┠䛻䛚䛔䛶᭷ព䛺ᕪ䛜ぢ䜙䜜䚸䛭䛾㛗ᮇ䜻䝱䞁 䝥䛾᭷ຠᛶ䛜♧䛥䜜䛯䚹^䠓䠅㻌

୍᪉䚸℧䜙䛿㏻Ꮫᆺ䜻䝱䞁䝥䛜Ꮚ䛹䜒䛾䛂⏕䛝䜛ຊ䛃䛻ཬ䜌䛩ᙳ㡪䛻䛴䛔䛶䚸䠥䠧䠮ホᐃ⏝⣬䜢⏝䛔䛶᳨

ウ䛧䛯䚹^䠔䠅㻌 䛭䛾⤖ᯝ䚸䜻䝱䞁䝥ึ᪥䛸᭱⤊᪥䜢ẚ㍑䛧䛯䛸䛣䜝᭷ព䛻ྥୖ䛧䚸䠍䞃᭶ᚋཬ䜃䠎䞃᭶ᚋ䜒䛭䛾 䛂⏕䛝䜛ຊ䛃䛿ᣢ⥆䛧䛶䛔䛯䛣䛸䛜᫂䜙䛛䛻䛺䛳䛯䚹䛧䛛䛧䛺䛜䜙䚸䛭䛾䛂⏕䛝䜛ຊ䛃䛾ᵓᡂせ⣲䛻䛹䛾䜘䛖䛺 ኚ໬䜔≉ᚩ䛜䛒䛳䛯䛾䛛ලయⓗ䛻䛿᫂䜙䛛䛻䛥䜜䛶䛔䛺䛔䚹㻌

䛭䜜䜖䛘䚸ᑠᏛᰯ㛗ᮇ⮬↛య㦂άື䛾ᐇ᪋ᮇ㛫୰䛾≉ᚩ䜢ぢฟ䛧䚸᪥䛜⤒䛴䛻䛴䜜䚸䛹䛾䜘䛖䛺ኚ㑄 䛜䛒䛳䛯䛾䛛䚸ฟ᮶஦䛻ᑐ䛩䜛ኚ໬䛿ఱ䛛䛒䛳䛯䛾䛛䛺䛹䚸ලయⓗ䛻ᐇ㊶䛸◊✲䜢↷䜙䛧ྜ䜟䛫䜛䛣䛸䛜 ᚲせ䛷䛒䜛䛸⪃䛘䛯䚹㻌

䛭䛣䛷䚸ᮏ◊✲䛷䛿ᑠᏛᰯ㛗ᮇ⮬↛య㦂άື䛾ཧຍ⪅䛜⏕䛝䜛ຊ䜢䛹䛾䜘䛖䛻ᙧᡂ䛧䛶䛔䛳䛯䛾䛛䚸 䛭䛾≉ᚩ䜢 㻝 㐌㛫䛾᫬⣔ิⓗ䛻᫂䜙䛛䛻䛧䚸䛛䛴ᛶᕪ䛻㛵䛧䛶䜒ゎ᫂䛧䚸䛭䛾⮬↛య㦂άື䛾ᚰ⌮Ꮫⓗ

䛺ᩍ⫱ຠᯝ䜢᳨ウ䛩䜛䛣䛸䜢┠ⓗ䛸䛧䛯䚹㻌

䠎䠊 ௬ㄝ㻌

㻌 䛂⏕䛝䜛ຊ䛃䠄䠥䠧䠮䠅ホᐃ⏝⣬⡆᫆∧䠎䠔㡯┠䛿䚸ᚰ⌮ⓗ♫఍ⓗ⬟ຊ䞉ᚨ⫱ⓗ⬟ຊ䞉㌟యⓗ⬟ຊ䛾䠏䛴䛾ୖ

఩ᑻᗘ䛛䜙ᡂ䜛䛜䚸ᮏ◊✲䛷䛿㌟యⓗ⬟ຊ䛛䜙䠐㡯┠䜢๐㝖䛧䛯䚹䛭䜜䛿䚸㔝እάື䛾ᐇ⩦䛺䛹䜢㏻䛧 䛶㌟యⓗ䛻䛭䛾ᢏ⬟䛜ྥୖ䛩䜛䛾䛿ᙜ↛䛾䛣䛸䛷䛒䜚䚸䛣䛣䛷䛿䛭䜜௨እ䛾ศ㔝䛷䛹䛾䜘䛖䛺ኚ໬䛜ぢ䜙 䜜䜛䛛䜢୰ᚰ䛻᳨ウ䛩䜛䛣䛸䜢ㄢ㢟䛸䛧䛯䛛䜙䛷䛒䜛䚹㻌

䠥䠧䠮䛾ᚰ⌮ⓗ♫఍ⓗ⬟ຊ䛸䛿䛭䛾㡯┠䜢ぢ䜛䛸䛂䛭䛾ሙ䛻䜅䛥䜟䛧䛔⾜ື䛜䛷䛝䜛䛃䛂๓ྥ䛝䛻䜒䛾䛤䛸 䜢⪃䛘䜙䜜䜛䛃䛺䛹䛾ෆᐜ䛜୰ᚰ䛸䛺䛳䛶䛚䜚䚸䛣䜜䛿⌧ᐇᑐฎ⬟ຊ䛷䛒䜛䛸ゝ䛔᥮䛘䜙䜜䜛䚹ᚨ⫱ⓗ⬟

ຊ䛷䛿䚸䛂ே䛾ᚰ䛾③䜏䛜ศ䛛䜛䛃䛂ே䛾䛯䜑䛻ఱ䛛䜢䛧䛶䛒䛢䜛䛾䛜ዲ䛝䛰䛃䛺䛹䛾ෆᐜ䛜୺䛷䛒䜚䚸䛣 䜜䛿௚⪅䜈䛾ཷᐜ䛷䛒䜛䛸ゝ䛘䜛䚹䜎䛯䚸䛂ከ䛟䛾ே䛻ዲ䛛䜜䛶䛔䜛䛃䛂⮬ศ䛾䛣䛸䛜኱ዲ䛝䛷䛒䜛䛃䛾㡯┠

䛛䜙䛿䚸⮬ศ䜢䜘䜚䜘䛔Ꮡᅾ䛸䛧䛶ㄆ▱䛧䛶䛔䜛䛣䛸䛛䜙⮬ᕫ⫯ᐃ䠄⮬ᑛ䠅䛸⌮ゎ䛷䛝䜛䚹㻌

ドキュメント内 茨城SCフォーラム第4号 (ページ 44-60)