6.4. LOGLINEAR (MANOVA) 121

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1 Φ Γ SPSS Dr. υ υ α α Θ α 2012

2 2 1. Γ SPSS Φ Γ SPSS Φ Γ Φ Pτ ΘHKH IAΓPAΦH Γ Γ Φ Γ Θ Γ Φ 2.1 Θ, Γ, Γ Θ Γ Φ Γ Γ Θ Γ Φ Γ Γ Γ Γ Γ I Θ 4.1 Γ Φ Γ (Independent samples t-tests) Γ Γ Γ (Paired samples t-tests) Γ (AστVA) Γ Γ υ α π α π υ α π α π 74

3 Γ Γ Γ Γ Γ Γ Γ ( Kruskal-Wallis) Γ Γ Γ 6.1. (CRτSS TABUδATIτσ) Γ Φ LOGLINEAR PEARSτσ SPEARεAσ Γ (PCA) (CA) (DA) (MANOVA) 121. Γ 125

4 4 1. Γ SPSS 19.0 α π α α SPSS (Statistical Package for the Social Sciences) α α απ α α α α πα α α π α π α α α υ α π. 1.1 Φ Γ SPSS SPSS υπ υ α α α: α (SPSS Data Editor), α α απ (SPSS Viewer). SPSS Data Editor α α φ α α, π α α α α π υ υ α α α υ. SPSS Data Editor απ α απ πα υ α: Data View ( α 1.1) α Variable View ( α 1.2). π υ α α π υ α α α υ α υ α α αυ, α υ π υ α α αυ. α Data View α Cases ( π ) α α α α υ α, α Variables ( α α ). Γ α πα α, υ α α α υ α 10 α, α α π α α α υ, α α α α α α α α π α ( α 1.3).

5 5 α 1.1. O SPSS Data Editor πα υ Data View α α α 1.2. SPSS Data Editor πα υ Variable View α φ π

6 6 α 1.3. O SPSS Data Editor (πα υ Data View) α α υ α υ 10 α SPSS Viewer α α απ ( α 1.4). α υ πα υ, Output, φα α α π π υ υ α α α απ α α. α 1.4. O SPSS Viewer α πα υ α απ

7 7 α (menu bar) SPSS Data Editor π α π : File, Edit, View, Data, Transform, Analyze, Direct Marketing, Graphs, Utilities, Add-ons, Window, Help. υπ υ α SPSS Viewer, π υ υπ υ π π α Insert α Format. π υ α π π υ α υ αυ π α : File: π α α υ α α (σew), α πα (τpen), α απ υ α α (Save), α υπ υ (Print),... Edit: π α π π υ α α υ α α υ α υ. View: π α π α υ α φ α α υ πα α υ α α π α. Data: π α π α α π υ α α α α. Transform: π α π α α π υ α α α. Analyze: α α π α α υ. Direct Marketing: φα α α π α. Graphs: υ αφ πα α. Utilities: α α α π. Γ α πα α, α π φ α α α α α. Add-ons: α π πα IBM ( α α - α υ υ SPSS) Window: π α α π πα υ. Help: φ φ α α. απ α υπ α α (toolbars), π α π φ α α π υ α α. 1.2 Φ Γ απ π α α Σ α φ α α α απσ υ α π Data View. α πα α α α 1.5, π υ α α π α απ υ απ α φ υ α α (documented

8 8 collection), α απ π α α α αυ υ α. φ φ υ, π π α α α α (hand arthritis), π π φ υ φυ π υ (lumbar vertebrae osteophytosis), υ υ α υ. α α αυ υ φ απ α α. π π α αφ υ α απ α φ υ Excel φ υ SPSS π α α α φ υ Excel, α φ α Ctrl+C α π α α φ υ SPSS Ctrl+V. α 1.5. α

9 9 1.3 α α π υ π υ α α (variable) α, π α αφ, α α α υ SPSS πα υ Data View. α α π, α (numeric) α α φα (string). α α α α α φ α α, α φα φ α α α, α α α υ α α φα υ, υ υα α α α π π. π π α α α : π (quantitative) α π α (qualitative/categorical) α. π έ α έ α π υ π α, π,, α α,. υ π πα υ α έ έ α φ ά α α ά α έ, π π υπ α. Γ α πα α (gr, kgr, ), α α ph. SPSS π α α α α α α (scale). Γ α, α π α α ί α α έ α ή α (interval) α α έ α α ία (ratio). H α αφ π α α αυ α α α α α α υ α π α α α α φ π α. υπ π π α α α α α α, υ α α 0 C α απ υ α α α α. Ό π α, α α υ α π υ π α π υ, α α υ (continuous) α α (discrete) α υ (discontinuous) α. υ α π α π υ π α π π α α, αφ α υ α υ π α π. α α α πα υ υ, υ α α, α υ υ α α α π υ α αυ. π έ α α α α α φ υ π α α υ π υ. SPSS α α α α αυ α α (nominal) α α α α (ordinal).

10 10 α (nominal) α α π α α υ α α υ. Γ α πα α, α α π υ φ α πα f ( υ α α) α m ( α ) α α. Θα π α α α f α m α π α υ α 1 α 2, α α. α π 1 α 2 α α α. α α α α (ordinal) α α π α α υπ υ α α. Γ α πα α, α α υ α 1.5 α α (arthritis) υ α φ (osteophytosis) α π α π υ ordinal. π α πα 1, 2, 3, 4, 5, 6, 1, 2 α 3. α π π α υ α α α α υ α π. α α 1. π ή α ώ ώ α ά α α ά ύ ό από ύπ α ώ π υ ά α. α α 2. Γ α α υ α α π υ α α α α α α α (α φα α -string) α απ α Σ α φ α α, α π π π α α υ π Variable View ( π πα α ). 1.4 Φ α, α φ π απ πα υ Data View : υ Variable View α π (Name) π π φα π υ υ α υ ( α ) Data View (π.. sex, bday, arthritis, osteophytosis, height, bmass). (Type) π υ π α. υ α αυ, α υ φα α α. αυ φα α α πα υ α υ π υ α π π α π υ π α. υ α υ π : Numeric, Comma, Dot, Scientific notation, Date, Dollar, Custom currency α String. α α SPSS απ υ

11 11 π υ Numeric α String π 6 π π π υ α. Comma α α α α α π α α α α α, π.. 5, Dot α α α α α π α α α α α α, π ,6. scientific notation α π π πα υ α α α, π α α α α Date π α α α υ. πα α π υ υ, α π υ Date α φ π α birthday, υ α π υ φ dd.mm.yyy, π φα α α 1.6. Dollar π α α α αφ α α α α υ., π π π Custom currency. α 1.6. φ π Width α υ π α α α π α α α. Decimals α α α α φ α α. Label ( ) π α υ α π αφ α. Values υ π φ α α α αυ α α. π π α None α αφ υ π α. α πα α α α, arthritis,

12 12 π α πα 1, 2, 3, 4, 5 α 6 α α π π α α, π υ α π α α υ α α α. Σ αυ π π, α α π φ, υ Values π υ α α arthritis α, π α πα υ α υ υ α π α Value π 1, Label π slight osteophytic formation α υ Add. φ α 1 = Οslight osteophytic formationπ α π α ( α 1.7- ). υ υ α 2 Value, moderate osteophytic formation Label α π Add. αυ π α π υ α υ α 1.8. α π υ α osteophytosis ( α 1.9). α 1.7. α α υ π υ π α υ α υ Value δabels α α arthritis

13 13 α 1.8. πα υ α υ Value Labels α α arthritis α 1.9. πα υ α υ Value Labels α α osteophytosis SPSS π π α α υπ υ. Γ Σ αυ π α υ ( πα α 0) α α υ α απ α. πα α α π υ 0 α απ α bmass π αυ π α υπ φυ bmass. απ α

14 14 Missing (Values) α α π α α π υ α π υ. υ α α υ π υ φα α α αυ. α πα υ α υ υ α α α π α α π υ υπ υ απ Discrete missing values υ π α π υ α αφ α α υ απ. π, π α υ α απ υ, π.. απ 0-10 α υ π α α, α π α α α π π (π , 333). α πα υ α υ Missing Values Columns α υ π π υ α α α (π α φ α π α π α ). Align α α α π Left (α ), Right ( ) α Center ( ). Measure α α α α α π (Scale) α (Nominal) α (Ordinal ). α φ π υ α 1.5 α α α α 1.11

15 15 α α φ π υ α 1.5 α α φ π υ α 1.5 α α : SPSS π α π υ α α.

16 Pτ ΘHKH Ή IAΓPAΦH υ α υ α α π π ( α ) α π π π υ υπ υ, π υ α π π α π υ α ά ω απ π υ υ α υ α α α απ Edit π υ Insert Case. α α υ α α π υ α απ π υ α υ α α α π υ Insert Case. Γ α α υ α α α ( ) α α π υ υπ υ, υ Σ α π π π υ α ά απ π υ υ α υ α α απ Edit π υ Insert Variable. α α π υ π υ α απ π υ α υ α α π υ Insert Variable. α α, α α α Edit Undo. Γ α α α υ α π π ( α ) α α ( ), π υ α αυ α π υ Clear. 1.6, α πα α, υ α α υ α α (m) α α α (cm). π Transform Compute Variable α υ πα υ α υ υ α 1.13, υ height (height),, π α height α π α Numeric Expression, α υ υ π α / α 100. Target Variable υ, heightm, α Type & Label α υ φ υ π α α υ height in m. ΟΚ α α α α π α υ α α. α α. α π α Compute Variable α π υ α α α υ υ α α α υπ υ α, α π π αυ α υ α φ υ Excel α α α αφ υ α α SPSS.

17 17 α πα υ α υ Compute Variable 1.7 υ α υ α ( α height) α : α απ 160 cm, α 161 Ν 170 cm, α 171 Ν 180 cm α α α απ 181 cm. α αυ απ υ 1, 2, 3 α 4, α α. Γ α α υ π α υ πα α π α: π Τransform Recode Into Different Variables α υ πα υ Recode into Different Variables. Σ αυ π υ α height α υ π α Numeric Variable Output Variable. π α Name π α α, heightcat, α π α Label π α α Σ αυ, height category. Change πα υ α υ α 1.14.

18 18 α πα υ α υ Recode into Different Variables αυ υ Old and New Values, π α α πα υ α υ, Recode into Different Variables: Old and New Values. υ π υ πα υ αυ α υ α α α υ α α α: 1. υ Range, LOWEST through value α π α π υ υπ απ υ α 160. π α New Value π 1, π α π α υ π Add. Σ αυ α π α Old New φ α Lowest thru υ υ Range α υ υ α 161 π α 170 π α απ Range. π α New Value π 2 α υ Add. 3. πα α α υ α α 180 α α 161 α 170, α α. π π α New Value π 3 α υ Add ( α 1.15). 4. υ Range, value through HIGHEST α π α π υ υπ απ υ α 181. New Value π 4 α υ Add.

19 19 5. υ α α α Continue α ΟΚ. heightcat φα α α π ( α 1.16). α α. π α π Into Same Variable α α Into Different Variable, α α α α α α α salary απ α, heightcat. Γ αυ α π π. α υ π υ Recode into Different Variables: Old and New Values

20 20 α α heightcat 1.8 πα α π υ υ, α arthritis α ordinal α πα 1, 2, 3, 4, 5, 6 π υ υπ υ π π α α απ 1 (slight osteophytic formation) 6 (ankylosis). α υ α π υ α π π α π α, π α α π π α 1 α 2, 3 α 4, α 6 α 7. α π π α υ α α : α α αφ Low Low Moderate Moderate High High

21 21 Γ α α π υ αυ πα α π α υ π α α π. π Τransform Recode Into Different Variables α υ πα υ Recode into Different Variables α υ Reset. π υ α arthritis, π α υ π α Numeric Variable Output Variable. π α Name π α α α, arthritis2, α π α Label π α α Σ αυ, arthritis levels. Change πα υ α υ α υ Old and New Values α υ π υ πα υ α υ Recode into Different Variables: Old and New Values : Old Value υ πα, New Value α α υ Add. α α α αυ πα α α α α π υ α υ α υ Continue α ΟΚ, π arthritis2 φα α α π. α α αυ α φ π α α α Value Labels α α α α α. α πα υ α υ Recode into Different Variables

22 22 α πα α π arthritis πα υ α υ Recode into Different Variables: Old and New Values α α. Ό α π πα α πα υ Recode into Different Variables α απα α α αφα υ πα π απ π α Old New. υ α α π υ π υ υ α αφα υ α υ Remove Reset. 1.9 α α α α α απ Data Sort Cases. πα υ Sort Cases π υ α υ α π α α α, α π υ α α α α υ α (Ascending) φ υ α (Descending) ( α 1.19). υ ΟΚ. α α α α α α α α α, υ π π ( α ) α α α α α α υ α α α π υ υ π. α α α α α π α α υ α α, π α π α

23 23 α, α π υ Sort Ascending Sort Descending. α πα υ α υ Sort Cases 1.10 Γ α α α αυ φα α α υ α υ α υπ α. Γ α πα α, υ α υ α α α υ α α α υ α 1.5. απ Data Select Cases α υ α πα υ α υ α υ If condition is satisfied α ο If πα υ π υ α π υ α sex α αφ υ π α π υ α. υ υ υ π = α π ΟfΠ. Θα π υ α υ α Continue α ΟΚ π α α π π π π υ sex = ΟfΠ. υ φα α απ α α π π sex = ΟmΠ υ α αφ, π φα α απ α αφ α π π sex = ΟmΠ α π π, π α υ α α α filter_$ α 0 α sex = ΟmΠ α 1 α sex = ΟfΠ. π αυ π α, α υ α π α π α π α α, αυ π α π π π υ sex = ΟfΠ.

24 24 α πα υ α υ Select Cases: If α α απ α π α υ α αφ π π sex = ΟmΠ.

25 25 Γ α α απ π υ π Select Cases π α υ απ Data Select Cases α πα υ α υ α υ All Cases Reset. π, α α α α υ α filter_$, αφ π α π υ α α πα υ π Delete Γ Φ SPSS πα υ α α υ α π αφ αφ πα α α απ α. Γ α πα α, υ α π α α α πα α π υ υ. Γ α π αυ, α υ π α Graphs Legacy Dialogs Scatter/Dot. π α α υ π υ α π υ Simple Scatter α υ Define, π φα α πα υ α υ υ α πα υ αυ υ α bmass α α υ π α απ π α Y Axis, α bmass α α π α π υ αφ α y. π υ α height π α X Axis α sex π α Set Markers by. α α α π α α X Axis, Y Axis α Set Markers by

26 26 πα υ αφ πα α υ α υ αυ αφ πα α απα φ π α α α α υ α, α α α α α υ α π υ α α. Γ α α φ π υ α αφ πα α υ π π, π α π α αφ πα α (Chart Editor) α υ α π π υ υ. Γ α α φ υ απ π α υ π α. Ό α α π α αφ πα α α υ Σ α απ υ α α α α, π α α α αυ α α α π α α υ π π α α υ α α,, α, α α α α φ, α π α α υ α α φ υ. φ π υ υ α α α π π α πα υ π Apply. π α υ Σ α π υ πα α α α αφ πα α, π α α α α α α α α υ π α α α α. π π α π υ φα α π α α φ π υ α α α α π, α α υ ( α 1.24). α Γ αφ πα α α body mass α height

27 27 α φ π αφ πα α α π απα α. π υ, π α αφ υ π υ π α φ α. αυ υ α α α αφα α. π π π α α α- π υ α α αυ υ π α. υ αφ πα α α α α π α φ α α Γ Γ α α α υ α απ υ α υ SPSS α υ α Windows α α α α α υ: υ π π υ π α υ File Open, α. α αφ πα υ π π α α υ α α υ Excel α υ SPSS. Γ α π αυ α υ π α File Open Data α πα υ α υ Open File π υ α π υ Look in φ α υ π α α α. Files of Type υ π υ α υ, α Excel (*.xls, *.xlsx, *.xlsm), α π υ α π υ υ α α υ. Οpen α πα υ υ α α υ Excel π α ( υ ), υ Read variable names from the first row of data. π π υ π φ α α α α

28 28 αφ υ SPSS α υ ΟΚ. π αυ φ α α π υ π α α π α αφ υ SPSS. α πα υ α υ Opening Excel Data Source Όπ υ α αφ, π υ Excel α αφ υ α α α υ SPSS Copy - Paste. α, π υ α π απ α φ α α υ Excel, υ Ctrl+C, αφ α αφ υ SPSS, υ π α υ φ υ α α υ SPSS α υ αφ Ctrl+V. α α α α α α φα ( α αφ απ SPSS Excel) Θ Γ α α απ υ α α υ SPSS υ File α, π υ Save as ( πο υ ω ) α υ π υ α α πα υ α υ π υ α φα. α, SPSS Data Editor, απ α π α.sav. α, α α απ, SPSS Viewer, απ α π α.spo. α α α υ SPSS π α απ υ α α α υ Excel. α α απ α α υ Excel α πα υ α υ Save Data As, π υ α α File Save as, π υ Excel 97 through 2003 (*.xls) Excel 2007 through 2010 (*.xlsx) π α Save as type. υ α υ α υ π α File name α υ Save. π α υ SPSS απ α π υ φ υ α α π υ α Data View.

29 29 2. Γ Φ H α π π α α υ π α α π α π -π α α α. π αφ α α α π υ α απ υ α υ π α απ α πα υ α α) α π αφ, ) π υ α ) αφ πα υ α. 2.1 Θ, Γ, Γ π υ α υπ. Γ α πα α, αφ α α α α α α α α α, π υ απ α απ α α α α α π υ υ. α α α υπ υ π υ π π υ α α α υ. Γ α πα α, α α α α π υ υ α α υπ π α π α α υ α α υ α, π α π υ υ α α α π α αυ α απ α α. π υ α υ α α ( α π υ π υ α α π α α α α α) α α υ α α. α, π α υ π, π π α υ α υ α α φ π υ α υ υπ υ π υ α. α αυ π π, α υ α π υ α απ υ π υ α απ αυ α υ, π α υ α α α υπ ( υ α α) α α π υ α α α ( υ α α). υ α πα α α π α α απ. α π π α α υ α α) υπ α α π υ, ) π υ α α απ α ) π π π α α α. α α α υ α α α α α. π α απ α π υ π υ, α α π α α π υ αυ α α α π π υ υ π υ. α α α π π υ υ π υ,

30 30 α α απ υ, α α υ α α α α υ π α α. 2.2 Θ Γ Φ α α α α α υ φ. α α α α π α α : α, α π α α α α ( α α 2.1). α α υ π φ π υ α υ α, α α α π υ π α π α α α α, α α α α α αφ α α α, α π υ α α α α α α απ π α. α α 2.1. α α α α (Mean) (Median) υφ (Mode) α (First quartile) o α (Third quartile) α α π α π (Variance) υπ απ (Standard deviation) υπ απ υ (Standard error of mean) (maximum) (Minimum) α α (Interquartile range) α α α α υ α υ α (Skewness) υ υ α (Kurtosis) (mean average value) υ α α απ π α α υ υ α α α απ : x = (x 1 + x x m )/m

31 Όπ υ x 1, x 2, Ζ, x m α α m υ α. Γ α πα α, υ α α α α α υ υ υ α, α α υ π π π υ α α 30 ( α m=30), x 1 α υ π υ α, x 2 υ υζ x 30 υ α α. α π α α (variance) α α π α απ υ α α απ : Var(x) = ( x x) 1 2 ( x 2 2 x) m 1 ( x m x) 2 α π α υ υ α π υ α. Γ α πα α, υ α π α υ α, α π π α α π α π π. α π π, α α πα α υ α. υπ απ (standard deviation) α α α α π α π φ απ απ x. (median) α Ο α απ α, α υ α α α υπ π α. υ α π υ, π π α α υ υ α α α υ α (απ π α ). Γ α πα α, α υ π α = {1.53, 1.65, 1.78, 1.84, 1.86}. α α α = {1.53, 1.65, 1.78, 1.80, 1.84, 1.86}, υπ α απ ( )/2=1.79, α π π απ - α α. π α απ α α (π.. α α υπ α α )., α π αφ π υ φα υ α α π α απ, π α π α π απ α α. υφ (mode) α α υ α Σ α α. Γ α πα α, α = {2, 3, 5, 3, 6, 2, 4, 3} υφ α 3., α (First, third quartile) α α α (Interquartile range): α α α α (quartiles). π α (Q 1 ) α υ α

32 32 α π α 25% υ α α αυ. α (Q 3 ) α υ α α π α 75% υ α α αυ. αφ Q 3 -Q 1 α α α. υπ απ υ (Standard error of mean) υπ α απ s s m = m π υ s α υπ απ α m υ α. (maximum) α υ α. Γ α πα α, α α α = {1.53, 1.65, 1.78, 1.84, 1.86} α (minimum) α υ α. Γ α πα α, πα απ α α υ α υ α (Skewness) α υ α (Kurtosis). Όπ α φ α 3, α α α α υ π α α α π υ π α α υ α. π υ π α α α α α α υ α υ α α υ α. υ α υ α υ α υ α 3 α υ υ α α 4. Ό α α 3 = 0 α α α α υ π, α α 3 < 0 α α α α π α α, α π α α υφ, α α 3 > 0 α α α α π α ( α 2.1). α 3 >0 α 3 =0 α 3 <0 α 2.1. Κα α ο αφο α υ α Ό α α 4 = 0 υφ α α αυ υπ α α α ( φ α 3). Ό α α 4 < 0 α α π α, α α 4 > 0 α α α υφ ( α 2.2).

33 33 α 4 <0 α 4 =0 α 4 >0 α 2.2. Κα α ο αφο υ α α α π αφ α υπ α α α α SPSS, π α πα α. 2.3 x 1, x 2, Ζ, x m α α x Σ α α. υ υ α x i φυ α i π υ π φ πα α α α x i α. v = v 1 + v 2 + Ζ + v m, fi α υ α x i. Γ α πα α, α ={2, 5, 3, 5, 8, 9, 6, 2, 5, 8, 7}. υ α 5 α : f 5 = 11 3 π φα α φ α 11. i Ό α π υ α α α υ α α x α υ, α π α π α π α π π ( α πα α, α α υ α, α α α α α υ α ), υ α α π π υ α. υ α α x min α x max α α α x α, α α x max - x min k υπ α α α π υ α α υπ υ υ α π υ α υ Σ αυ. π α αυ α υ α. Γ α πα α, υ α υπα α α α : ={28, 36, 22, 41, 27, 50, 32, 29, 42, 29, 25, 38, 36, 45, 27, 29, 32, 39, 47, 33, 53, 33, 31, 40, 20, 34, 37, 29, 33, 27, 39, 37, 44, 26, 43, 26, 36, 34, 49, 36, 26, 31, 28, 59, 30, 28, 30, 34, 28, 24}

34 34 α υ α α π υ 8. α αυ πα α α 20 α α 59. π φα α πα απ π υ : 5., π α 5, π 8 α α α : 20-25, 25-30, 30-35, 35-40, 40-45, 45-50, 50-55, π α υ απ 20 24, απ υ π, α α υ α α 2.2. α α 2.2. α α υ υ πα α α υ α Γ Γ α α υπ υ SPSS α α α α scale α (π, υ α ) α υ α α α: Analyze Descriptive Statistics Descriptives α πα υ α υ Descriptives π υ α π υ υ α α α υ α α αυ α. Options α πα υ Descriptives: Options π υ π α π υ α α π υ υ α υπ ( α 2.3-α ). α α αυ α α. α α α π α υπ υ α α α Analyze Descriptive Statistics Frequencies. α α α αυ υ Statistics α π υ α α π υ υ απ πα υ Frequencies: Statistics ( α 2.3- ).

35 35 α 2.3. α πα υ α α υ Desciptives: Options α Frequencies:Statistics α α α υπ, υπ απ, υπ απ υ υ (S.. mean), α α Quartiles,, υφ, α α υ α α height υ α υ osteological data.sav. υ α osteological data.sav α α υ α α α Analyze Descriptive Statistics Frequencies, π π π υ υ α υπ υ α πα υ α υ Frequencies: Statistics. πα υ Frequencies π υ α υ α salary α. π, απ π π Display frequency tables α Statistics π υ π π υ υ α υπ. υ α υ Continue α ΟΚ. α απ α α π υ πα υ α α 2.4. α αυ α α quartiles α α percentiles 25 α 75.

36 36 α 2.4. α υ απ 2.5 Γ Ό α α α α nominal ordinal π α υπ α πα απ α α. Σ αυ π π υπ α υ, α α π φ αφ α. π α π υ α υ α : Analyze Descriptive Statistics Frequencies. π υ α π υ, π π Display frequency tables α Statistics απ π π. α α α υπ υ α arthritis α osteological data.sav.

37 37 α α osteological data.sav α υ α α α Analyze Descriptive Statistics Frequencies α πα υ α υ Frequencies υ α hand arthritis α. π π Display frequency tables α Statistics απ π π. α απ π υ πα υ α α 2.5. π π α α απ πα α α πα α 70% α α α α α π α π π α, ankylosis 4%. α 2.5. α υ απ 2.6 Θ Γ Φ π υ α π αφ πα α α πα υ α α. π α α α α) α α α (bar charts), ) υ α α α (pie charts), ) α α (histograms) α ) α α (boxplots). π π αφ πα α

38 38 π α α α α π (scale), υ α π α υ π α (nominal α ordinal α ). π αφ α α α α (histograms), α α α (bars) α υ α α α (pie). αφ αυ πα α φα α α απ α πα υ α υ Frequencies υ Charts α π υ α φ α ( α 2.6). α α, αφ πα α π α υ α α α Graphs Bar (Pie or Histogram), π α πα α πα α. α 2.6. πα υ α υ Frequencies:Charts α) α α α (bar chart) α α α α α π α α υ α π α x. α π α απ α α α υ α α α α α υ α υ υ. α α α α α α α hand arthritis υ α υ osteological data.sav.

39 39 α) Γ α α α α υ υ α α α α arthritis α υ α α α Graphs Legacy Dialogs Bar α πα υ α υ π υ Simple α υ Define ( α 2.7). πα υ α υ π υ α π υ α arthritis π α υ π α Category Axis. π απ Bars Represent π υ α y. υ π υ N of cases % of cases. π π π υ α α α υ α π π π υ α απα, π π α υ α α υ π π π π υ α απα. ΟΚ πα υ α 2.8. α 2.7. πα υ α υ Bar Charts

40 40 α 2.8. α α α α hand arthritis ) υ α α (piechart) α α αυ α α υ. α φ α υ α α α α π υ α υ υ. Γ α υ φ α α α α α. π Graphs Legacy Dialogs Pie πα υ α υ π υ Summaries of groups of cases α υ Define. πα υ α υ υ π α π α π π π, α αφ υ α π υ υ α υ π α Define Slices by α π υ υ N of cases. α 2.9 α υ α α π υ α α lumbar vertebrae osteophytosis.

41 41 α 2.9. υ α α α lumbar vertebrae osteophytosis ) α α (histogram) α α υ α α, π υ α π α α υ α α π υ υ α. π α α α υ α α α 5 α 25 α α υ α. α π α α α α α α α υ α. SPSS α υπ α απ π α α αυ α υ α υ α. α α data.sav. α α α α body mass υ α υ osteological α α π α α α αφ scale α, α α α α α α υ αφ α α nominal α ordinal α. Γ α α α α υ υ α α α body mass α υ α α α Graphs Legacy Dialogs Histogram. πα υ α υ π υ α body mass α π π Display normal curve ( α 2.10). H α α α α α α π,

42 42 α α π α α α, α α π φ α. ΟΚ πα υ α α υ α α πα υ α υ Histogram α α α α body mass

43 43 α α α α υ π π π α α α α απ υφ. Ό α α α, α π α α π α α. α 2.11 πα α α α φα α α α υ α α. υ π α φ α α π υ υ ( υ) α α υ α. π α υ α 2.11 α α α πα απ α π α α. Γ α α υ π π π α α υ α α, αυ φ π α Histogram υ α sex π Panel by Rows ( α 2.12). αυ π α υ α α, α α υ α α α υ α ( α 2.13). α πα υ α υ Histogram

44 44 α α α α body mass α α φ α α π α υπ υ αφ α α, α π υ υ υ α α α α α. ) Θ α α (boxplot) α α απα α απ α α, α υ υ α α π υ ( α 2.14). υ υ α Q 1 (π α ) α π Q 3 ( α ). α απα α α α υ α α υ υ. α α α π υ π α α : α) α υ α, ) α υ α π υ α απ Q (Q 3 - Q 1 ) α υ α π υ α α απ Q 1 Ν 1.5(Q 3 - Q 1 ), ) α υ α π υ α απ Q 3 + 3(Q 3 - Q 1 ) α υ α π υ α α απ Q 1-3(Q 3 - Q 1 ).

45 45 α Θ ογ α α α α π α α α αφ scale α. α α αυ, α υ υ α, α α υ α α α α body mass φ. Γ α α α α α υ υ α υ α α α Graphs Legacy Dialogs oxplot. πα υ α υ oxplot π υ Simple, Summaries for groups of cases α υ Define ( α 2.15). υ π υ πα υ α υ Define Simple Boxplot: Summaries for Groups of Cases π α 2.16 α πα. Θα π υ α α α υ α α πα υ α υ Boxplot

46 46 α πα υ α υ Define Simple Boxplot: Summaries for Groups of Cases α Θ o α α α body mass α φ

47 47 α α αφ π α φ. Όπ α α α, υ α υ α α. α α α α α α α, α αυ α υ α α α. α αυ α α α α π α φ α α.

48 Γ Γ Όπ α αφ, α α α υ π π π α α α α. υ α α π α π, π απ π π, α α α α υ π π π π α α α α α υ α α. α α α 3.1 α 3.2. α α α α α α α α αφ. π α α α α υ π, α α. αυ π π α α α π α απ αφ α α. α 3.1. α α α 3000 π υ α υ α α α =5 α =1

49 49 α 3.2. α α α 3000 π υ απ υ απ α α α υ α α α α π υ π α απ π υ α α υ α απ α υ υ α α υ, π υ α 3.1. α π υ αυ α α α α α π υ α υ α α α. υ α α αφ υ απ πα απ, α α α απ απ α α απ α α υ α α α α α π φ α απ υ α α α, π α α υ α 3.2. π π α υπ α υ π α α π φ α α. υ αυ α υ πυ α π α α α α. 3.2 π α π π α α α π υ υ φ α α α. π αυ π α : υ α α

50 50 α α Poisson α α α υπ α α α α α Student t α α - α α Fisher F π α α α, α υ α α υ π α α α π υ α υ υ α υ π υ π α α υ πυ α π α α α π π, π α πα α: Ό α α π α α π υ υ α απ α α π α p α q=1-p, α α, α π υ π π α υ υ α α. α π α α α υ α α Poisson. α π α α α, υ π υ α απ α α υ, α υ α α α. 3.3 Γ α α α απα α α π υ α α α., α α π π α α π α α α α α α υ π α. α α α α Kolmogorov- Smirnov α Shapiro-Wilk. SPSS υπ α α αυ α υπ α π α α α υ α α α υ α α υ α α α. π α α αυ SPSS υ α Sig. υ α Sig. α απ 0.05 α α α α α υ α. α α υ Sig. α υ α π π φ α. Γ α α α SPSS α υ π α: Analyze Descriptive Statistics Explore α πα υ α υ υ α π υ π α Dependent List. π α α Explore

51 51 απ φα Kolmogorov-Smirnov α Shapiro-Wilk π φ α π α υ α: π Statistics υ υ α : Descriptives: π α υ α α α. Outliers: π 5 α α 5. π ο Plots π α α α υ υ α α α α: Boxplots, istograms α Normality plots with tests. π Options π α απ π : Exclude cases listwise: υ υπ π α π π π υ α αυ α υ α π υ υπ υ Dependent δist α Factor δist. Exclude cases pairwise: υ υπ π α π π π υ α υ α α π υ υπ Dependent List. α α α α α α height υ α υ osteological data.sav. υ α osteological data.sav α α υ α α α Analyze Descriptive Statistics Explore. πα υ α υ υ α height π α Dependent List α υ υ π Plots. π α α υ π υ φα α υ π None π Boxplots, απ π π Stem-and-leaf π Descriptive α π υ Normality plots with tests ( α 3.3).

52 52 α 3.3. α υ α υ Explore: Plots π α απ α α π υ πα υ αφ π α α Tests of Normality ( α α 3.1) α α α Q-Q π υ α o α 3.4. π α α 3.1 π α Sig. (significance) α π α α α υ α απ α α υ α α υ α α α. Γ, π α αφ, α Sig. α α απ 0.05 α α α α α. Ά α α π υ υ α υ α α α υ Sig.(Kolmogorov- Smirnov) =0.2 α Sig.(Shapiro-Wilk) = α α 3.1. π α α υ α α Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. height,102 50,200 *,961 50,095 a. Lilliefors Significance Correction *. This is a lower bound of the true significance. υ π α α π π α απ α α Normal Q-Q (quartilequartile) plot υ α 3.4. Γ α α α α α α π π

53 53 α α αυ α α α α π υ α. α α αυ υ α α α π υ υ. α α α α υ α α α υ α Q-Q α α α απ π α α α. Γ αυ α υ α α απ α α υ π α α Tests of Normality. α 3.4. α α Q-Q α α α α α height απ α απ υ α α υ α υ α. υ α υ α α φ υ α α. υ α α α Analyze Descriptive Statistics Explore α υ π υ πα υ α υ π υ α π α 3.5. α υ α απ α α υ α α 3.2, απ π π π α υ υ φ υ α υ α α α.

54 54 α 3.5. α υ α υ Explore α α 3.2. π α α υ α α Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk sex Statistic df Sig. Statistic df Sig. height f,110 21,200 *,941 21,231 m,121 29,200 *,945 29,139 a. Lilliefors Significance Correction *. This is a lower bound of the true significance. 3.4 Ό α υ α α α αφ α x υπ απ s α α. π υ α αφ α α π υ α υ π α α α x π υ α υπ απ, α α υπ απ υ π υ απ π π α α. Σ αυ π π π α α α α π. υ α α P% α π (confidence interval) α πα α υ υ π υ, α ( 1, 2 ) α π α α α α υπ π α α P%. υ π α α %

55 55 φ α : = 100(1-α). Σ αυ π π α φ υ φ α, α π α α πα υ π υ α α απ α π. π α α α α π π υ π α υ α α α 95% α π. α α ( 1, 2 ) α π α α α α υπ υ π υ, α π α α, π α α 95%. α αυ υπ α α α π υ α υ α α α, π π φ α πα α πα α. α α α υπ 95% α π α α height υ α υ osteological data.sav. υ α osteological data.sav α α υ α α α Analyze Descriptive Statistics Explore. υ π υ πα υ α υ π α 3.5 α πα Statistics. πα υ α υ π υ α π υ Descriptives, α α π, α υ α α υ α α π, 95% ( α 3.6). α Continue α ΟΚ. α α α α π πα υ α π α α Descriptives ( α α 3.3). α 3.6. π ογ α α ο π ο

56 56 α α 3.3. π α α π αφ α α π Descriptives sex Statistic Std. Error height f Mean 161,52 2,246 95% Confidence Lower Bound 156,84 Interval for Mean Upper Bound 166,21 5% Trimmed Mean 160,98 Median 162,00 Variance 105,962 Std. Deviation 10,294 Minimum 148 Maximum 185 Range 37 Interquartile Range 15 Skewness,707,501 Kurtosis,208,972 m Mean 175,90 1,698 95% Confidence Lower Bound 172,42 Interval for Mean Upper Bound 179,37 5% Trimmed Mean 176,03 Median 178,00 Variance 83,596 Std. Deviation 9,143 Minimum 159 Maximum 190 Range 31 Interquartile Range 15 Skewness -,274,434 Kurtosis -,989,845 α α π α α 95% π α α α υ α α α α 156,84 α 166,21 cm, α α α 172,42 α 179,37 cm.

57 Γ α α α α α π υ π π α α α α π. Γ α α α υ α α α π α υ Graphs Legacy Dialogs Error Bar. υ α υ υ α α α π υ π υ πα α α. πα υ α υ π υ α υ Simple, π υ Summaries for groups of cases ( α 3.7) α πα Define. υ π υ πα υ α υ Define simple Error Bar: Summaries of Groups of Cases, π α 3.8. α α π υ πα υ α α 3.9 α α π π α π φ α α α α α π π υ υπ α π α. α 3.7. πα υ α υ Error Bar

58 58 α 3.8. α α υ Define simple Error Bar: Summaries of Groups of Cases α 3.9. α α 95% α α α π α height

59 59 4. Γ Θ 4.1 Γ π α α υ α π υ απ φ π α α α υ α υ α. απ φ αυ α α απ φ. Γ α α υ α απ φ α απα α α υ υπ. α π α υπ, π υ α υπ (null hypothesis) α υ α 0, α αφ π α α α φ α υ α α φ α α, α υπ υ α α αφ α. α α α υπ 0 υ α 1. α α α απ υ α υπ π υ α α, υ α φ α π υ. α α α α α υπ, υ α φ α π υ. υ υ α π πα α π υ α φ α π υ αυ υ π α α α υ α φ α π υ. π π α α υ π α α α υ φ α π υ α α α αυ υ π υ. υ π π α α (significant level) π α α π α α α υ φ α π υ α υ α α υπ. π α α αυ υ α α α π υ υ π α α = 0.05 α = υ α π α α α απ υ α υπ α απ 5% α α = 0.05 α απ 1% α α = Θα π π α α α π π υ α πα υ απ φ π α, απ υ απ υ α υπ. π α απα π απ υπ. α απ π υ 0 υπ α π α α απ α% α α. α α υ

60 60 α α α, α 0 π π α α α, π α υ υ α υ. π, α απ π υ 0 π π α α α%, α φ α α υ π υ 0 α π α α 1-α/100. SPSS α α υπ p-value, α π α α α υ απ π α υπ. υ π α υ π π π α α α ( ), : p < α 0 απ π α p > α 0 απ π α SPSS p α απ Sig. (Significance). Ό α υπ π π υ α α υ α α υ α α α. υ π π π υ π π α α α α α, π π φ πα απ. α πα α α α αυ υ φα α υ α υπ υ αυ α α α α α α υ α α α πα υ υ α απ απ αυ. 4.2 Φ Γ (Independent samples t-tests) π π π, υ α υ υ α α α α α α υπ α α αφ α υ. Γ α πα α, π α υ α υ υ α α α α. Γ α π αυ υ υ. α α π υ π υ 10 α α 8 υ α α α απ α α 1 α 2, α α, α 4.1 ( cm). α α α πα υ υ α α απ π π α α 0.05.

61 61 Γ α α π υ α α α υ α αυ α α SPSS α αφ υ α, π υ υ samples. π α, π υ υ groups, π υ α 1 α 2 α α α υ α α α, π φα α α 4.1. α 4.1. α πα α α Γ α α π υ SPSS πα απ, α υ α α α Analyze Compare Means Independent-Samples T Test, πα υ π υ α αφ υ α Samples π α Test Variable(s), α Groups Grouping Variable α υ Define Groups. πα υ υ 1 Group 1 α

62 62 2 Group 2. Continue α ΟΚ πα υ υ α 4.1 α 4.2. α α 4.1. Γ π αφ α α α α α α Group Statistics groups N Mean Std. Deviation Std. Error Mean samples ,350 3,0252, ,400 2,6753,9459 α α 4.2. α απ α υ Equal variances assumed Equal variances not assumed Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference,567,462-1,502 16,153-2,0500 1,3648-1,524 15,793,147-2,0500 1,3453 α α α α π Levene p = > 0.05 π υ 0 απ π α. υ π υπ α α αφ α π. αυ απ α πα απ π α α π α (Equal variances assumed). υπ α α αφ α π, α α α απ α α α (Equal variances not assumed) πα απ π α α. π α απ α α π α πα υ α υπ p = > 0.05 π υ 0 απ π α.

63 63 π π π α α 0,05 υπ : υπ υ α α αφ α. 4.3 Γ Γ Γ (Paired samples t-tests) α α α υ α α υπ α π α α α α. Γ α πα α, π α α υ α 4.2 α 8 αφ α α αφ υ. αυ α υ α. Σ αυ π π α π υ α α α υπ α α αφ α. υπ α α αφ, α π π αφ α α α α. α α Σ α φ 8 αφ α α. α απ α α π υ π α α α α 4.2. α α υ α α α απ α α. αφ υ α α φ α α υ SPSS, π φα α α 4.2 π υ α υ α α α Method1 α Method2. υ α α α Analyze Compare Means Paired-Samples T Test. πα υ π υ α υ α α αφ υ π α Paired Variables ( α 4.3). ΟΚ πα υ υ π α απ. π α α π υ υ α αφ α Paired Samples Tests ( α α 4.3). α α p = > 0.05 π υ α 0 απ π α. υ π υπ υ α α αφ α α α υ α α απ α α.

64 64 α 4.2. α υ α 4.3. α α υ Paired-samples T test

65 65 α α 4.3. π α α α υ Paired Samples Test Method1 - Method2 Paired Differences 95% Confidence Std. Std. Error Interval of the Difference Sig. Mean Deviation Mean Lower Upper t df (2-tailed) -,1250,2188,0773 -,3079,0579-1,616 7, Γ (ANOVA) Γ (One-way ANOVA) υ α α αφ α α α. π π π υ α υ α υπ υ α α αφ α π. υ π υ α π υ α α υ α π (Analysis of Variance - ANOVA). α υπ υ π πα α α υ, π α α υ π π : -πα α α υ α π (One-way ANOVA) α -πα α α υ α π (Two-way ANOVA). π υπ π π : α υ α π α π. αυ υπ π π α α α υ πα α πα α. Γ -πα α α υ α π υ n α α (cases) m α (variables) α α. α α 4.4 α α υ α -πα α α υ α π. α α 4.4. υ α -πα α α υ α π α 1 x 11 x x 1m α 2 x 21 x x 2m α n x n1 x n2... x nm

66 66 Γ α α α π π φα υ π π α π α π π :. π π α υπ υ α α αφ α π. α α π π α υπ α α π (Homogeneity of variance). α π α α π α α, π α SPSS π Levene. υπ α α π, π α π υ πα α ANOVA, π α π φ α.. α α α π π α α υ α α α. απ απ α α α π υ α απ α α υ. υπ υ α απ φα υ πα α ANOVA. Γ α α φα υ απ One-way ANOVA SPSS α α α α π α α, α,, π υ α 1, 2, 3, Ζ α α α υ α α α. απ Analyze Compare Means One-Way ANOVA α υ πα υ One-Way ANOVA α αφ υ α π α Dependent List, α υ α 1, 2, 3, Ζ Factor α απ Options π υ Homogeneity of variance test α α υ α α π. υ Continue α ΟΚ. α α αφ π π π α. α απ α α α α α 4.5. α α α υπ υ α α αφ α υ π π α α α = 0,05. αφ υ α πα απ α φ α α υ SPSS, π φα α α 4.4, Height α Groups. Γ α α φα υ α υ α π α υ π α π υ π α αφ α. α, απ Analyze Compare Means One-Way ANOVA α υ πα υ One-Way ANOVA α αφ υ α Height π α Dependent List, α Groups Factor α απ Options π υ

67 67 Homogeneity of variance test. Continue α ΟΚ πα υ α απ α α 4.6 α 4.7. α α 4.5. Ύ cm α π π απ αφ α αφ π π α Ύ cm Γ α 4.4. α πα α α

68 68 α α 4.6. π α α υ α α π Test of Homogeneity of Variances height Levene Statistic df1 df2 Sig α α 4.7. π α α α height ANOVA Sum of Squares df Mean Square F Sig. Between Groups Within Groups Total π π α α Test of Homogeneity of Variances πα α α α π Levene p = > 0.05 π υ 0 απ π α. υ π υπ α α αφ α π α α ANOVA π α φα. α α π α p-value ANOVA p = > υ π π π α α α = 0.05 υπ υ α α αφ α υ α π. α α υπ υ π α α α 195, 180, 170, 185 α 190. υ π α α π π υ Σ αυ π π ; α α υ α απ α α π πα απ, πα υ α α 4.8 απ π φα α π π α α α = 0.05 (α α π π α α α = 0.01) υπ υ α α αφ α υ π (p = < 0.01).

69 69 α α 4.8. π α α α ANOVA height2 Sum of Squares df Mean Square F Sig. Between Groups Within Groups Total Γ α α α π π α αφ, α α : πα υ One-Way ANOVA υ Post Hoc α π υ Tukey, φ υ α α π. α α απ α α α π υ α α α 4.9. π π α α αυ α π π υ α α αφ α groups 1 α 2 (p = < 0.05) α 1 α 3 (p = 0.01 < 0.05). α, υπ υ α α αφ α groups 2 α 3. α 1 =, 2 = α 3 = Γ. α α 4.9. π α α π υ υ Multiple Comparisons height2 Tukey HSD Mean 95% Confidence Interval (I) groups (J) groups Difference (I-J) Std. Error Sig. Lower Bound Upper Bound * * * * *. The mean difference is significant at the 0.05 level Γ (Two-way ANOVA) Όπ υ α αφ, -πα α α υ α π (Two-way ANOVA) α α υπ π π : α υ α π α α υ α π.

70 πα α α υ α π υ α π α απ πα, α -πα α α υ α π π υ υ α πα α. α α π α αφ α (,, α IV) π υ α αφ π : ( ), ( ) π, α (C). π α α π π π υ α α α απ α α π υ φ α α α α α α α π α α α π α π υ α π α π υ α υ. α α α πα α α C IV αφ υ α α υ α α 4.10 α φ α α υ SPSS, π φα α α 4.5. α αυ α Dimensions φ π υ α, Date π υ α Location π α. φα 1, 2, 3, 4 Date α π υ I, II, III, IV α 1, 2, 3 Location π, α C. Γ α α φα υ α -πα α υ α π α υ π α Analyze General Linear Model Univariate. πα υ π υ α αφ υ α Dimensions, Date, Location α π α α Dependent Variable α Fixed Factor(s) π φα α α 4.6. υ Model α πα υ α υ π υ α υ α υ : π υ Custom, αφ υ

71 71 α Date α Location π α Model α π υ All 2-way, π φα α α 4.7. υ υ Continue α πα υ Univariate υ Options, π υ π υ Homogeneity tests. Continue α ΟΚ πα υ α απ α α υ α α α 4.5. α πα α α SPSS

72 72 α 4.6. πα υ Univariate α 4.7. πα υ Univariate: Model

73 73 α α π α α υ Dependent Variable:Dimensions Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 23,750 a 5 4,750 11,400,005 Intercept 35316, , ,200,000 Date 20, ,750 16,200,003 Location 3, ,750 4,200,072 Error 2,500 6,417 Total 35343, Corrected Total 26, a. R Squared =,905 (Adjusted R Squared =,825) p = < 0.05 π π α α α = 0.05 π α υ πα α χ ο π ο ο α α α. α, π α π α α α α α α α (p = > 0.05). π πα α α π α υ πα α χ ο π ο ο α α α, π SPSS α υ α π π υπ υ α αφ. Όπ α απ One-Way ANOVA, αυ α Post Hoc πα υ Univariate. πα υ π υ α αφ υ α Date π α Post Hoc Tests for α π υ Tukey, φ υ α α π. α α απ α α α π υ α α α 4.12, απ π π π α α αφ υπ υ α π Ν, Ν V, II Ν III α II - IV. α α αφ Ν V α II - IV.

74 74 α α π α α π υ υ Multiple Comparisons Dimensions Tukey HSD Mean 95% Confidence Interval (I) Date (J) Date Difference (I-J) Std. Error Sig. Lower Bound Upper Bound I II,00,527 1,000-1,82 1,82 III -2,00 *,527,034-3,82 -,18 IV -3,00 *,527,005-4,82-1,18 II I,00,527 1,000-1,82 1,82 III -2,00 *,527,034-3,82 -,18 IV -3,00 *,527,005-4,82-1,18 III I 2,00 *,527,034,18 3,82 II 2,00 *,527,034,18 3,82 IV -1,00,527,321-2,82,82 IV I 3,00 *,527,005 1,18 4,82 II 3,00 *,527,005 1,18 4,82 III 1,00,527,321 -,82 2,82 Based on observed means. The error term is Mean Square(Error) =.417. *. The mean difference is significant at the.05 level π π π π α υπ α α π α α πα α απ α υ π α α α α α π α π απ αυ π υ α α α α. Σ αυ π π π π α π υ υ α π α π. Γ α α φα αυ α υ απα α α υπ υ π απ α υ π υ υ υ πα α. α α Σ α π α α π α α α α υ ph α π υ α υ 24 φ φ α α απ α α π υ α

75 75 α α α α π α α α α υ ph α π υ υ α υ. α α α πα α α / 0 C ph = 5 ph = 6 ph = αφ υ α α α φ α α υ SPSS, π φα α α 4.8, π υ α φ πυ α, α α α ph α υ α α. α 4.8. α υ πα α α SPSS

76 76 Γ α α α α υ α α α υ π α Analyze General Linear Model Univariate. πα υ π υ α αφ υ α π α Dependent Variable α α ph Fixed Factor(s). Options π υ Homogeneity tests, α Model π υ Full Factorial. α απ α α π υ πα υ α α α α υ π α α π υ π π υ απ π α α αυ α α α υ α: (i) υπ α α π α α α α π υ υ α υ (p = > 0.05). ( ) π α υ ph α α α (p = < 0.05). (iii) π α α π α α α α α ph (p = < 0.05). υ α α απ υ α υ ph α α απ α α α α φα. α α π α α ANOVA Dependent Variable:B Tests of Between-Subjects Effects Source Type III Sum of Squares df Mean Square F Sig. Corrected Model a Intercept T ph T * ph Error Total Corrected Total a. R Squared =.918 (Adjusted R Squared =.842)

77 ΓEσIKA Όπ α αφ, α α φα υ π υ φα α υ α απα α α α α α υ α α α. π υ π π υ α α α α πα α. Ό α α α α υ α α α α υπ α φα υ πα α. -πα α α α π υπ υ πα α π φ π υ α α α π. α αυ α π φ π υ πα υ α απ α πα α. π, α πα α, π Post Hoc α -πα α α υ α π πα α π α υ πα α. 5.2 Γ Γ αυ π α α υ π α α α α υ α απ υ α π α απ π υ. α α υ πα α Independent samples t-test. α α α α α α υ α υ independent samples t-tests.sav πα α. υ α independent samples t-tests.sav α α υ α α α Analyze Nonparametric Tests Legacy Dialogs 2 Independent Samples. αφ υ α Samples π α Test Variable List, α Groups Grouping Variable α υ Define Groups. πα υ υ 1 Group 1 α 2 Group 2. Continue α ΟΚ πα υ α α 5.1. α α p = ( 0.068) > 0.05 π υ 0 απ π α. υπ 0 α α α α π α

78 78 απ π υ. υ π, απ α αυ π π π π α α 0.05 υπ υ α α αφ α. α α 5.1. π α α Test Statistics b samples Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed).062 Exact Sig. [2*(1-tailed Sig.)].068 a a. Not corrected for ties. b. Grouping Variable: groups 5.3 Γ Γ Γ α α α α α α πα α Paired samples t-test. α α α α α α υ α υ paired samples t-tests.sav πα α. υ α paired samples t-tests.sav α α υ α α α Analyze Nonparametric Tests Legacy Dialogs 2 Related Samples. πα υ π υ α υ α Method1 α Method2 α αφ υ π α Test Pair(s) ( α 5.1). π υ Wilcoxon (α α π ) Test Type α ΟΚ πα υ α α 5.2. α α p = > 0.05 π υ α 0 απ π α. υ π υπ υ α α αφ α α υ π υ α α απ α α.

79 Test Statistics b Method2-79 α 5.1. α α υ Two-Related-Samples Tests α α 5.2. π α α Method1 Z -1,436 a Asymp. Sig. (2-,151 tailed) a. Based on negative ranks. b. Wilcoxon Signed Ranks Test Γ ( KRUSKAδ-WALLIS) α α α α α α π πα α One-way ANOVA. α α α α α α υ α υ one-way ANOVA.sav πα α.

80 80 υ α one-way ANOVA.sav α α υ π α Analyze Nonparametric Tests Legacy Dialogs Κ Independent Samples, α υ πα υ Tests for Several Independent Samples α αφ υ α Height π α Test Variable List α α Groups Grouping Variable. υ Define Groups α πα υ υ 1 Minimum α 3 Maximum. Continue α ΟΚ πα υ α α 5.3. α α p = > υ π π π α α α = 0.05 υπ υ α α αφ α υ α απ π. α α 5.3. π α α υ Test Statistics a,b height Chi-Square 2,800 df 2 Asymp. Sig.,247 a. Kruskal Wallis Test b. Grouping Variable: groups υ α height2 πα α πα υ πα α π α α απ απ π φα α π π α α α = 0.05 (α α π π α α α = 0.01) υπ υ α α αφ α υ α. α α 5.4. π α α υ Test Statistics a,b height2 Chi-Square 8,089 df 2 Asymp. Sig.,018 a. Kruskal Wallis Test b. Grouping Variable: groups

81 81 α α 1. Σ αυ π π π α π υ π π φ α π α αφ. Γ α α π υ αυ π α υ α α α ( α 5.2). α α αφ π α α υ π υ α α υπ π. α α 2. πα απ α α α α α Kruskal - Wallis. α 5.2. Θ α α α α π α α αφ π α Γ α α α α υ πα α Two-way ANOVA. α α α α α α υ α υ two-way ANOVA.sav πα α.

82 82 α 5.3. α πα α α α α υ α φ α α α αφ υ α α υ α υ two-way ANOVA.sav π φα α α 5.3. υ α α α Analyze Nonparametric Tests Legacy Dialogs Κ Related Samples α πα υ π υ α αφ υ α A, B, C π α Test Variables. π υ Friedman (α α π ) Test Type α ΟΚ πα υ α α 5.5. α α p = > 0.05 π υ α 0 απ π α. υ π υπ υ α α αφ α α αυ α α. φ α α π α, π α υ πα α π α α α α. α α 5.5. π α α Test Statistics a N 4 Chi-Square 4,667 df 2 Asymp. Sig.,097 a. Friedman Test

83 83 Γ α α π α π υ υ α α α : π Data Transpose πα υ π υ α αφ υ α,, C π α Variable(s) α υ ΟΚ. α α α α φ α Σ α φ α α π υ α αυ α α ( α 5.4). αυ φ πα α α υ α α α Analyze Nonparametric Tests Κ Related Samples α α α π α Test Variables. α απ α α π υ πα υ α α α 5.6. α 5.4. α α υ α υ α α 5.6. π α α Test Statistics a N 3 Chi-Square 8,143 df 3 Asymp. Sig.,043 a. Friedman Test α α α π α χ ο π ου α α α π π α α α = 0.05 (p = < 0.05). α α π α απ πα α

84 84 α υ α α υ π υ π φ α υπ υ αφ, π υ π π α α α α αφ α. Γ α α α π π (,, α IV) υπ υ α α αφ, α α υ υ α α α αυ., α υ α α α Graphs Legacy Dialogs oxplot. πα υ α υ oxplot π υ Simple α Summaries of separate variables π α α α α αφ ( α ) ( α 5.5). πα υ α υ Define Simple Boxplot: Summaries of Separate Variables αφ υ α,,, IV π α Boxes Represent α πα ΟΚ. Θα π υ α α α υ α 5.6. α 5.5. π α α α υ α α α α α αφ φα α α π ( α ) α ( α IV).

85 85 α 5.6. Θ α α φ π υ I, II, III, IV

86 86 6. Γ Γ α π υ α α π α φ α α αφ π α. φ α απα α α α α υ α α. α α α π π υ α π π α π α α α α. 6.1 (CROSS TABULATION) Ό α υ α π α α, π αυ υ α α α, α α α α π υπ α φ υ α α α. αυ, π υ α α nominal ordinal α π α α α α crosstabs (cross tabulation). α α α π φ υ α α α υ φυ π υ α osteological data.sav. υ α osteological data.sav α α υ α α α Analyze Descriptive Statistics Crosstabs. υ π α Row(s) α sex α Column(s) α osteophytosis. υ Cells α π υ α Observed, Expected, Row, Column α Total. υ Continue α ΟΚ. π α α απ α α α 6.1. π α α αυ Count α π π π υ υπ υ α π α α α Expected Count π π π υ α α α α α α α υ α α. Γ α πα α, α α υπ υ 4 υ α eburnation, α α α φ πα α α α 4.6. Γ πα α α α υ πα α α π υ υ φ α α π π π.

87 87 υ α α α α α α π α α υ π α υ φ υ α α α α. α α 6.1. π α α φ υ α α α υ φυ π υ sex * lumbar vertebrae osteophytosis Crosstabulation lumbar vertebrae osteophytosis lipping pitting eburnation Total sex f Count Expected Count 10,9 5,5 4,6 21,0 % within sex 52,4% 28,6% 19,0% 100,0% % within lumbar vertebrae osteophytosis 42,3% 46,2% 36,4% 42,0% % of Total 22,0% 12,0% 8,0% 42,0% m Count Expected Count 15,1 7,5 6,4 29,0 % within sex 51,7% 24,1% 24,1% 100,0% % within lumbar vertebrae osteophytosis 57,7% 53,8% 63,6% 58,0% % of Total 30,0% 14,0% 14,0% 58,0% Total Count Expected Count 26,0 13,0 11,0 50,0 % within sex 52,0% 26,0% 22,0% 100,0% % within lumbar vertebrae osteophytosis 100,0% 100,0% 100,0% 100,0% % of Total 52,0% 26,0% 22,0% 100,0% α π αυ α α α α Nominal Ordinal π α α 2 (chi square test). Γ α α υ αυ, πα υ α υ Crosstabs π υ α απ Analyze Descriptive Statistics Crosstabs, υ Statistics α π υ Chi-square. αυ υπ ( 0 ) α α α α α υ α p-value α Assymp. Sig.

88 88 Γ α α φ υ α π π υ α α υ φυ π υ πα υ α α 6.2. α α p = > 0.05 α υ π π α απ υ υπ. υ α φα α α υπ α α π α α φ υ α π π υ α α υ φυ π υ. α α 6.2. π α α υ 2 test α φ υ α π π υ α α υ φυ π υ Chi-Square Tests Value df Asymp. Sig. (2-sided) Pearson Chi-Square,237 a 2,888 Likelihood Ratio,238 2,888 N of Valid Cases 50 a. 1 cells (16,7%) have expected count less than 5. The minimum expected count is 4,62. υ π α α α α υ α υ π α φ υ α α α α α ( α α 6.3). α α 6.3. π α α υ 2 test α π α φ υ α π π υ α α α α Chi-Square Tests Value df Asymp. Sig. (2-sided) Pearson Chi-Square 2,381 a 5,794 Likelihood Ratio 2,418 5,789 N of Valid Cases 50 a. 8 cells (66,7%) have expected count less than 5. The minimum expected count is,84.

89 Γ Φ α αφ πα υ υ αφ πα α π υ α π α α α. πα α α α φ υ α π π υ α α υ φυ π υ αφ πα α α α υ α α π υ Display clustered bar chart πα υ α υ Crosstabs π υ α α α α Analyze Descriptive Statistics Crosstabs. υ α α 6.1 α α π π υ φ α α π π α π. α 6.1. α α α φ υ α π π υ α α υ φυ π υ α α φ α αυ α α απ Graphs Legacy Dialogs Bar. πα υ α υ π υ Clustered α Summaries for groups of cases α υ Define. πα υ α υ π υ α αφ υ α sex π α Category Axis α α osteophytosis π α Define clusters by. ΟΚ πα υ π α 6.1. α α α φ αυ υ α π π α π υ Stacked α α Clustered ( α 6.2) π υ υ α π φ αυ υ α 6.1.

90 90 α 6.2. υ α α α φ υ α π π υ α α υ φυ π υ 6.4 LOGLINEAR υ Loglinear π α α α υ υ α π α α. π α αυ α υ α απ πα α πα α. α α υ α υ υ α α ( α- α) α πα υ α π φα α (πα απ ) α α α απ α α (0 Ν 1). υ α αφ α α υ α υπ αφ π α α α αυ. α α α α α α 6.3 α 6.4. α υ Loglinear π πα α π υ α π υ π π υ υ α π α α. απ

91 91 α α α (independence model). πα α π υ υ υ α α α απ α α m, b α s. αυ π π α α απ : α 6.3. α α loglinear analysis α 6.4. α α 6.3

92 92 ln(fr ijk ) = c 0 + c 1 m i + c 2 b j + c 3 s k π υ c 0, c 1, c 2, c 3 α π α πα π υ υπ α SPSS α m i, b j, s k α α π υ α πα α π υ υ α πα υ 0 α 1. α π α π φ α π α α. Γ α αυ, υ απ (saturated model) ln(fr ijk ) = c 0 + c 1 m i + c 2 b j + c 3 s k + c 4 mb ij + c 5 ms ik + c 6 bs jk + c 7 mbs ijk α απα φ υ α α - α υ α α υ α α α υ π υ π π υ υ α π α α, α 6.3. α mb ij, ms ik, bs jk, mbs ijk α α π (interactions) α φ υ α π α α m i - b j, m i - s k, b j - s k α m i - b j - s k, α α. Γ α α φα υ α υ Loglinear SPSS, απ α α υ π Data Weight cases. π π Weight cases by α αφ υ α frequency π α α υ Frequency Variable ( α 6.5). π αυ π α α α α α α α frequency α υ. α 6.5. α υ α υ Weight Cases

93 93 υ α α υ π α: Analyze Loglinear Model Selection. π υ α π α π α υ α υ α αφ υ π α Factors, π φα α α 6.6. υ υ π Define Range α α α υ α π υ α ( α 6.7). πα α π υ α α υ π 0 α 1. α 6.6. α α υ Model Selection Loglinear Analysis α 6.7 α π α

94 94 υ α υ υ π Options α π π Association table α απ α α α υ α π α α υ ( 2 ) α α. Continue α OK πα υ π π α, απ υ π υ αφ πα υ υ πα α : α α 6.4. π α α Goodness-of-Fit Tests Goodness-of-Fit Tests Chi-Square df Sig. Likelihood Ratio Pearson π υ π α Goodness-of-Fit tests. π αυ π α α α 6.4 α α αφ α. π α α αυ Chi-Square α 0 α αυ α π π φ απ υ α α α α υ. υ φα α α απ π π α α, Cell counts and Residuals, π υ πα α α (Observed) υ α π π (Expected) αυ α. α α K-Way and Higher-Order Effects α α π υ π α απ α υ απ α π υ α α απ α α. αυ π α α υ π α K-way Effects. α α π α α ( =1) α α α (Sig.=0.002). π, α π α α (motifs-burnish, motifs-site, burnishsite) π α (Sig.=0). α, α π α α (motifs-site-burnish) α α α π α (Sig.=0.225) α υ π υ α α αυ α π α απα φ.

95 95 α α 6.5. α α K-Way and Higher-Order Effects K-Way and Higher-Order Effects Likelihood Ratio Pearson Chi- Chi- Number of K df Square Sig. Square Sig. Iterations K-way and ,655,000 51,000,000 0 Higher Order ,868,000 36,450,000 2 Effects a 3 1 1,476,224 1,473,225 4 K-way Effects b ,787,008 14,550, ,393,000 34,977, ,476,224 1,473,225 0 a. Tests that k-way and higher order effects are zero. b. Tests that k-way effects are zero. α α 6.6. α α Partial associations Partial Associations Effect df Partial Chi- Square Sig. Number of Iterations m*b 1 33,275,000 2 m*s 1,671,413 2 b*s 1 3,681,055 2 m 1 4,307,038 2 b 1 7,468,006 2 s 1,012,913 2 π π α α (Partial associations) α α π π α α υ φ α αφ αφα π α motifssite-burnish. α α α α site (s) α α π, motifs-site (m*s), burnish-site (b*s), α α α. υ π π α α α υ υ π α α υπ αφ π α α α α α π υ α.

96 Σ α α π υ π α α α φ (x, y), α απα α α π υ π υ α π φ. H α α α αυ α πα (regression) α α α α π π π α α α α α α π α α απ α απ., π α π π αφ π α α α ω αχί ω ραγώ ω α α α π π υ π α απ α α (x i, y i ) α α π α α α υπ π α. υπ π (residual) α αφ α π α α α y α x. α α α α 7.1 α α υ υ υ α α π mm. α αφ πα α weeks - mm α α α π α α 50 α 55 mm, α α. α α 7.1. α υ υ υ α α π mm mm weeks mm weeks αφ υ α α Σ α φ α α υ SPSS α απ φα υ π α α α α α α π α α.

97 97 α α π υ α α α π α αυ π υ α υ α α α αυ π υ π υ π α α υ π α α α. π π α, φ αφ α α π υ α π υ υ υ υ α α, α α α υ α α α α α. π α, απ Analyze Regression Curve Estimation υ α weeks π α Variable(s) α α mm Independent Variable, π φα α α 7.1. π, π υ Display ANOVA table, Linear, Plot models α Include constant in equation. υ α α π α π α α υ α α x = 0 α y = 0. ΟΚ πα υ α π α α αφ πα α υ α 7.2. π υ π α αφ α α 7.2, π υ α πα α. α 7.1. α α υ Curve estimation

98 98 α α 7.2. υ π α Coefficients Unstandardized Coefficients Standardized Coefficients B Std. Error Beta t Sig. mm,511,035,973 14,541,000 (Constant) 4,420 2,215 1,996,069 α 7.2. αφ πα α weeks Ν mm υ α α α. π π α α π π υ α (y = a + bx) α : y = x H υπ απ α a α α b α 0.035, α υ α = α b = υ α α α α α α α α, a b, α α α. π Sig. α α απ α α α α π α α α. α α π α πα υ υ α 7.1 α π α Include constant

99 99 in equation. Γ α α α α α α π α απα φ απ, α υπ υ υ α πα α. Γ α α π υ α α π α α 50 α 55 mm, α α, απ υ π : 4,42 + 0,511*50 = 29, ,42 + 0,511*55 = 32,525 32,5 α α α α 7.3 α α α π α α. α αφ πα α υ π α α α α α 1150 α 1250 mm 2. α α 7.3. α π α α Thousands Tooth-size Thousands Tooth-size years ago (mm 2 ) years ago (mm 2 ) Θα π π α α α π υ α α. π α α α υ α 1150 α 1250 mm 2 α π υ α α tooth-size α α, years. υ α, α υ α α π α αυ π πα α, αφ π υ Quadratic α απ π Linear πα υ Curve Estimation. ΟΚ πα υ α 7.3 α α α 7.4.

100 100 α 7.3. αφ πα α years Ν tooth size α π α α α α 7.4. υ α π Coefficients Unstandardized Coefficients Standardized Coefficients B Std. Error Beta t Sig. tooth size -,859,184-5,520-4,677,001 tooth size ** 2,000,000 6,479 5,490,000 (Constant) 435, ,157 4,023,002 υ α π y = a + bx + cx 2 π α α Coefficients α : α = (Constant) = , b = toothsize = α c = toothsize**2 = ( SPSS ** α α ).

101 101 π π α π υ. toothsize**2 = α α α α α (Sig.=0,000) α α 0 α υπ υ φ α π υ φα α. Γ α π αυ π υ π α α α αφ υ α φ υ Excel. α υ π 0,000 α toothsize**2 απ υ αυ α 0, π α π π α αυ υ α α α α toothsize = -0,858616, φ α π υ υπ πα α. Γ α α π υ α 1150 α 1250 mm 2, π αυ υ π υ π α πα υ π α : 435,137 Ν 0,858616* , * = 9,26 435,137 Ν 0,858616* , * = 25,3 π α α α α α πα απ π φα α απ α π υ -0,859 α α 0,858616, πα υ 435,137 Ν 0,859* , * = 24,83 α υ 0 α α 0, α π υ 435,137 Ν 0,859*1250 = -638,6 α α πα απ α. α α π α y υ π υ α α απ αφ α α απ ( 1 ) α ( 2 ) α α υ φ υ α ( 3 ) α ( 4 ) α α υ α α φ α α α υ α α 26. α π α, α υ y = a 0 + a a a a 4 4

102 102 α α 7.5. α π α x υ π υ α α απ αφ απ α 1, 2, 3, 4. y αφ υ α α SPSS α υ α α 7.5. υ π α: Analyze Regression Linear α υ α y π α Dependent α α T 1, T 2, T 3 α T 4 Independent(s). π Options π υ Include constant in equation α Method π υ π υ α π α υπ α υ. Ό α π υ Enter π α α υπ α, π π π υ υ α a 0, a 1, a 2, a 3, a 4. π υ Backward π α α α υπ α α α α αφα α- α α α. π Forward π α α π α α α α π υ α α π υ α υ α

103 103 α. α α α α α α π α α π α α υ α α..., π Stepwise α υ υα Backward α Forward. Γ Stepwise, Forward α Backward π α α α π υ υ α α υ, Enter α. υ υ α υ π α απ α, π α α α π υ α α. α απ αυ α φυ α α υ υ υ. πα α π υ υ π υ Enter, πα υ α α 7.6, Backward α α 7.7. α α α π α α υ Backward π α α α α απ α. α α φυ π α φ α : y = Model 1 α α 7.6. π α α Enter (Constant) T1 T2 T3 T4 a. Dependent Variable: x Unstandardized Coeff icients Coefficients a Standardized Coeff icients B Std. Error Beta t Sig

104 104 Model α α 7.7. π α α Backward (Constant) T1 T2 T3 T4 (Constant) T1 T3 T4 (Constant) T1 T3 a. Dependent Variable: x Unstandardized Coeff icients Coefficients a Standardized Coeff icients B Std. Error Beta t Sig α α π υ α α πα α α α α α π α υ (correlation) α. α α π π α υ α υ α α α. α α α α α PEARSON SPEARMAN Γ α α υ α α, x α y, α, υπ υ υ υ Pearson, r. υ r πα α απ Ν1 1. υ r α υ α α x αυ, y α α α α φ. r = 0 α πα υ α r α α α α αυ, αυ α. Θα π π π α υ Pearson π α α α α α υ α α α. α υ α α α, υπ υ υ Spearman,, π υ π πα α απ Ν1 1, α α πα α υ.

105 105 α α α α α υπ υ α α height α body mass υ α υ osteological data.sav. υ α osteological data.sav α υ α α. Θα π π α υ υ α height α απ α α υ α α α. α α α body mass. π αυ α α α απ π α πα α υ α. α υ Analyze Descriptive Statistics Explore, πα υ α υ υ α α, height α body mass, π α Dependent List α υ υ π Plots. π α α υ π υ φα α υ π None π Boxplots, απ π π Stem-andleaf π Descriptive α π υ Normality plots with tests. π π α α απ, α α 7.8, πα α π α α α υ α α α. α α 7.8. π α α υ α α Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. height,098 48,200 *,963 48,134 body mass,116 48,111,965 48,167 a. Lilliefors Significance Correction *. This is a lower bound of the true significance. π π α π υ υ Pearson α π φα υ Spearman π υ υπ α π. υ α α υ α α α Analyze Correlate Bivariate. πα υ π υ α αφ υ α height α body mass π α Variables α π υ Correlation Coefficients α α Pearson α Spearman. π π υ (α α default) α Flag significant correlations. π αυ π α α α α α α

106 106 π π α α απ. ΟΚ πα υ α α 7.9. π υ υπ υ α α α υ υ (r = 0,863 α = 0,878). α α 7.9. π α α υ υ Pearson ( π ) α Spearman ( ) Correlations height body mass height Pearson Correlation 1,863 ** Sig. (2-tailed),000 N body mass Pearson Correlation,863 ** 1 Sig. (2-tailed),000 N **. Correlation is significant at the 0.01 level (2-tailed). Correlations height body mass Spearman's height Correlation Coefficient 1,000,878 ** rho Sig. (2-tailed).,000 N body mass Correlation Coefficient,878 ** 1,000 Sig. (2-tailed),000. N **. Correlation is significant at the 0.01 level (2-tailed) (Partial correlation) υ (Partial correlation) π α π υ α υ α π α α, υ υ π α α α. υ α, υ α α υ α α π α α α α α. α π α α υ, α (dichotomous), π α πα α φ ( α - υ α α), υ υα υ α α.

107 107 α α υ α α α π α α φ α α α α ( m 2 ) α α αφ π π υ π α α π α α α α. α α α α αφ α α 7.4, π υ α area πα α α υ, α fill φ π αφ π α α α α floor π α π α. α α υ α α fill α floor. α 7.4. α πα α α α υ υ αφ πα α fill Ν floor, πα α υπ α υ α α αυ α ( α 7.5). Ό υ π φ α α υ α α α α πα υ αυ, υ α π φ α α υ α π αφ π. Γ α α π α π φ α α (area) υ α fill α floor α α.

108 y = 0.679x R² = floor fill α 7.5. α α α α floor α fill φ α υ α α φ υ SPSS π φα α α 7.4, α υ π α: Analyze Correlate Partial. πα υ α υ π υ φα α αφ υ α fill α floor π α Variables α α area π α Controlling for, π φα α α 7.6. υ α υ υ π Options α π π Zero-order correlations, π φα α α 7.7. π αυ α α α απ α α υ υ α π α α. α, π α α απ α πα υ α, απ π α α area υ υ α fill α area, α υ Pearson α απ υ α fill Ν floor, fill - area α floor - area. ΟΚ πα υ α απ α α υ α α 7.10, π α α α. π α υ α απ α α απ υ α α α π α α area υ α fill α floor.

109 109 α 7.6. α υ α υ α Partial correlations α 7.7. α υ α υ Options α απ α α υ α υ α α α fill α floor (r=0.758 α p=0.011) α π α α α fill - area (r=0.809 α p=0.005) α floor - area (r=0.929 α p=0). υ α α α fill α floor α π α π φ α α α α π π α 7.5., α π α π φ α α, α α υ υ α fill α floor α α α

110 110 α area, υ υ υ α α (r=0.032) α π π αυ πα α α α α (p=0.934). α α α υ υπ α α, πα α α fill α floor πα υ α υ α. α α π α α zero-order α partial correlations Correlations Control Variables fill floor area -none- a fill Correlation 1,000,758,809 Significance (2-tailed),011,005 df floor Correlation,758 1,000,929 Significance (2-tailed),011,000 df area Correlation,809,929 1,000 Significance (2-tailed),005,000 df area fill Correlation 1,000,032 Significance (2-tailed),934 df 0 7 floor Correlation,032 1,000 Significance (2-tailed),934 df 7 0 a. Cells contain zero-order (Pearson) correlations.

111 Γ υ υ υ π α α υ α υ υ α υπ υ πα, α π α αυ. Γ α πα α, υ α α π υπ α α αφ α α α α α α απ α π α α α α π α α υ α, π, υ υ, α α. α αυ απ π πα α υ α (Multivariate Analysis). π α α π α α υ υ : α) υ υ υ (Principal Component Analysis - PCA), ) υ (Cluster Analysis Ν CA), ) α υ (Discriminant Analysis - DA) α ) υ α π α (Multivariate Analysis of Variance Ν MANOVA). Γ α φα PCA α CA απα α α α πα α φ π υ α α α. α, φα DA α MANOVA π π υ α α PRINCIPAL COMPONENT ANALYSIS (PCA) Γ α α π υ α υ α Σ α π α α υπ υ (clusters) α π π α α υ α υ π α α. α α π υ αυ φ υ α α ( α α ) α απ α α αφ πα α α α α α π α

112 112 π υ α α αυ π α. α α PC1 π α υ α. φ υ α α, PC2, π υ α PC1 α π φ υ, π α α PC1, α αυ α α α α α π π υ υ. αυ υ α π π. π π αυ π υ α α α. υ υ π α α α υ α PCs α α π υ α. α α 6 π υ π α α υ α 8.1 α α απ α α α υ α α π υ α αφ π,, C. α α υ π α α π υ. α 8.1. π α α α Al, Fe, εg, Si, Ca απ π,, C φ α α υ SPSS υ π α Analyze Dimension Reduction Factor α πα υ π υ α υ α Al, Fe, εg, Si, Ca π α

113 113 Variables. Rotation π υ π φ Varimax α Extraction π υ Principal Components, Correlation Matrix α Eigenvalues over 1 α α π υ υ π υ υ α απ 1 α π υ υ π α π α., απ Scores π υ Save as variables. αυ π PC1, PC2 απ α φ α α υ FAC1_1, FAC2_1. ΟΚ υ α αυ α α SPSS Data Editor FAC1_1, FAC2_1 π π α υ α PC1 α PC2. Γ α α αφ α απ α α α υ α α α Graphs Legacy Dialogs Scatter/Dot α π πα υ π υ α π υ Simple Scatter α υ υ Define. πα υ αφ υ α REGR factor score 1 π α X Axis, α REGR factor score 2 Y Axis α α Area Set Markers by. αυ π π,,, C, α αφ. ΟΚ πα υ ( απ α φ π ) α 8.2. α 8.2. α α απ (PC1 vs. PC2) α α α α π α υ α α (cluster), α α π α C α α α. α α π α υ π υ π α π α αφ απ αυ π α C α π π α α α α υ α υ π α C,

114 114 υ π α α. Σ αφ υ α υ π α C α π α υ α α α α υ. α α 1. α π α α υ α α υ π απ φ α α, FAC1_1, FAC2_1, FAC3_1, Ζ α α 2. α α απ υ (clusters) υ αφ υ π φ. α α α- φα υ α απ Rotation π υ π υ α α π φ (None) υ υ, Quartimax, Equamax, Promax CLUSTER ANALYSIS (CA) α υ π α υ π υ α υ α α α (clusters) πα. H υ α π α α π α α α α α φ α π α π α α αυ α. π υ α υ π α α α α, αυ α α α α. α α α 8.3 α 7 π υ α α υ α α π π π υ. αυ π α α α π α υ α π υ.

115 115 α 8.3 α α υ π α Analyze Classify Hierarchical Cluster α πα υ π υ α αφ υ α D1 Ν D7 π α Variable(s). Plots π υ Dendogram α υ Continue α ΟΚ. α α π υ πα υ α α 8.4. α 8.4. α α π υ