6.4. LOGLINEAR 90 8.5 (MANOVA) 121

Φ Γ SPSS Dr. υ υ α α Θ α 2012

2 1. Γ SPSS 19.0 1.1 Φ Γ SPSS 4 1.2 Φ Γ 7 1.3 9 1.4 Φ 10 1.5 Pτ ΘHKH IAΓPAΦH 16 1.6 16 1.7 17 1.8 20 1.9 22 1.10 Γ 23 1.11 Γ Φ 25 1.12 Γ 27 1.13 Θ 28 2. Γ Φ 2.1 Θ, Γ, Γ 29 2.2 Θ Γ Φ 30 2.3 33 2.4 Γ 34 2.5 Γ 36 2.6 Θ Γ Φ 37 3. - 3.1 Γ Γ 48 3.2 49 3.3 Γ 50 3.4 54 3.5 Γ 57 4. Γ I Θ 4.1 Γ 59 4.2 Φ Γ (Independent samples t-tests) 60 4.3 Γ Γ Γ (Paired samples t-tests) 63 4.4 Γ (AστVA) 65 4.4.1 Γ 65 4.4.2 Γ 69 4.4.2.1 υ α π α π 70 4.4.2.2 υ α π α π 74

3 5. 5.1 Γ 77 5.2 Γ Γ 77 5.3 Γ Γ Γ 78 5.4 79 5.4.1 Γ ( Kruskal-Wallis) 79 5.4.2 - Γ 81 6. Γ Γ 6.1. (CRτSS TABUδATIτσ) 86 6.2 2 87 6.3. Γ Φ 89 6.4. LOGLINEAR 90 7. - 7.1 95 7.2 104 7.2.1 PEARSτσ SPEARεAσ 104 7.2.2 106 8. 8.1 Γ 111 8.2 (PCA) 111 8.3 (CA) 114 8.4. (DA) 118 8.5 (MANOVA) 121. Γ 125

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 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 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 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 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 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 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 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.. 5,012.6. Dot α α α α α π α α α α α α, π.. 5.012,6. scientific notation α π π πα υ α α α, π.. 9.12 2 α α 912 9.12-2 α α 0.0912. 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 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 1.7-. 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 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 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 1.10. 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.. 0-10, 333). α 1.10. 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 1.12. φ π υ α 1.5 α α α α 1.11

15 α 1.11. α φ π υ α 1.5 α 1.12. α φ π υ α 1.5 α α : SPSS π α π υ α α.

16 1.5 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 α 1.13. 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 α 1.14. 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 160 1. 2. 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 2 171 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 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 1.15. υ π υ Recode into Different Variables: Old and New Values

20 α 1.16. 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 1 2 3 4 5 6 1 1 2 2 3 3 Low Low Moderate Moderate High High

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 1.17. 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 1.18. 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 1.17. πα υ α υ Recode into Different Variables

22 α 1.18. 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 α, α π υ Sort Ascending Sort Descending. α 1.19. 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 1.20. 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 1.21. 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 α 1.20. πα υ α υ Select Cases: If α 1.21. α απ α π α υ α αφ π π sex = ΟmΠ.

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. 1.11 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 1.22. 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 1.22. α α π α α X Axis, Y Axis α Set Markers by

26 πα υ αφ πα α υ α 1.23. 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 1.23. Γ αφ πα α α body mass α height

27 α 1.24. 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. 1.12 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 1.25. 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 αφ υ SPSS α υ ΟΚ. π αυ φ α α π υ π α α π α αφ υ SPSS. α 1.25. 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). 1.13 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 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 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 2 +... + 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 1.78. α α = {1.53, 1.65, 1.78, 1.80, 1.84, 1.86}, υπ α απ (1.78 + 1.80)/2=1.79, α π π απ - α α. π α απ α α (π.. α α υπ α α 1.35 2.10)., 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 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 1.86. (minimum) α υ α. Γ α πα α, πα απ α α 1.53. 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 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 α υ α α π υ 8. α αυ πα α α 20 α α 59. π φα α 59 20 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, 55-60. π α υ απ 20 24, απ 25 29... υ π, α α υ α α 2.2. α α 2.2. α α υ υ πα α α 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 υ α 3 15 12 9 5 3 2 1 2.4 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 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 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 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 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 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 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 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 α α π α α α, α α π φ α. ΟΚ πα υ α α υ α 2.11. α 2.10. πα υ α υ Histogram α 2.11. α α α body mass

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 2.12. πα υ α υ Histogram

44 α 2.13. 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 3 + 1.5(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 α 2.14. 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 2.17. α 2.15. πα υ α υ Boxplot

46 α 2.16. πα υ α υ Define Simple Boxplot: Summaries for Groups of Cases α 2.17. Θ o α α α body mass α φ

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 3. - 3.1 Γ Γ Όπ α αφ, α α α υ π π π α α α α. υ α α π α π, π απ 1000. 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 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 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 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 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) = 0.095. 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 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 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 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 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 3.5 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 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 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 = 0.01. υ α π α α α απ υ α υπ α απ 5% α α = 0.05 α απ 1% α α = 0.01. 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 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 (0.05 0.01), : 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 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 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 1 10 31,350 3,0252,9566 2 8 33,400 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 = 0.462 > 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 = 0.153 > 0.05 π υ 0 απ π α.

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 = 0.150 > 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 α 4.2. α υ α 4.3. α α υ Paired-samples T test

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 7,150 4.4 Γ (ANOVA) 4.4.1 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 12... x 1m α 2 x 21 x 22... x 2m...... α n x n1 x n2... x nm

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 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 178 158 148 170 139 168 153 147 165 178 Γ 135 175 173 155 153 α 4.4. α πα α α

68 α α 4.6. π α α υ α α π Test of Homogeneity of Variances height Levene Statistic df1 df2 Sig. 1.157 2 12.347 α α 4.7. π α α α height ANOVA Sum of Squares df Mean Square F Sig. Between Groups 384.533 2 192.267 1.415.281 Within Groups 1630.800 12 135.900 Total 2015.333 14 π π α α Test of Homogeneity of Variances πα α α α π Levene p = 0.347 > 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 = 0.281 > 0.05. 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 = 0.009 < 0.01).

69 α α 4.8. π α α α ANOVA height2 Sum of Squares df Mean Square F Sig. Between Groups 2140.133 2 1070.067 7.143.009 Within Groups 1797.600 12 149.800 Total 3937.733 14 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 = 0.038 < 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 1 2 21.80000 * 7.74080.038 1.1486 42.4514 3 27.80000 * 7.74080.010 7.1486 48.4514 2 1-21.80000 * 7.74080.038-42.4514-1.1486 3 6.00000 7.74080.725-14.6514 26.6514 3 1-27.80000 * 7.74080.010-48.4514-7.1486 2-6.00000 7.74080.725-26.6514 14.6514 *. The mean difference is significant at the 0.05 level. 4.4.2 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 4.4.2.1 -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 4.10. α α α π α α α π α π υ α π α π υ α υ. α α 4.10. α πα α α C 53 53 53 54 53 52 56 54 55 IV 57 56 55 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 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 4.11. α 4.5. α πα α α SPSS

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

73 α α 4.11. 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 35316,750 1 35316,750 84760,200,000 Date 20,250 3 6,750 16,200,003 Location 3,500 2 1,750 4,200,072 Error 2,500 6,417 Total 35343,000 12 Corrected Total 26,250 11 a. R Squared =,905 (Adjusted R Squared =,825) p = 0.003 < 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 = 0.072 > 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 α α 4.12. 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. 4.4.2.2 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 α α 4.13. α α π α α α α υ ph α π υ υ α υ. α α 4.13. α πα α α / 0 C ph = 5 ph = 6 ph = 7 25 25 30 30 35 35 40 40 9 11 13 17 18 22 22 28 18 20 23 27 27 33 20 24 36 44 27 33 23 27 7 13 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 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 4.14. 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 = 0.065 > 0.05). ( ) π α υ ph α α α (p = 0.001 < 0.05). (iii) π α α π α α α α α ph (p = 0.000 < 0.05). υ α α απ υ α υ ph α α απ α α α α φα. α α 4.14. 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 1649.833 a 11 149.985 12.161.000 Intercept 12240.167 1 12240.167 992.446.000 T 116.500 3 38.833 3.149.065 ph 330.333 2 165.167 13.392.001 T * ph 1203.000 6 200.500 16.257.000 Error 148.000 12 12.333 Total 14038.000 24 Corrected Total 1797.833 23 a. R Squared =.918 (Adjusted R Squared =.842)

77 5. 5.1 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 = 0.062 ( 0.068) > 0.05 π υ 0 απ π α. υπ 0 α α α α π α

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 19.000 Wilcoxon W 74.000 Z -1.870 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 = 0.151 > 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.

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 5.4 5.4.1 - 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 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 = 0.247 > 0.05. 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 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 5.4.2 - 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 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 = 0.097 > 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 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 = 0.043 < 0.05). α α π α απ πα α

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 α 5.6. Θ α α φ π υ I, II, III, IV

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 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 11 6 4 21 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 15 7 7 29 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 26 13 11 50 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% 6.2 2 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 Γ α α φ υ α π π υ α α υ φυ π υ πα υ α α 6.2. α α p = 0.888 > 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 6.3 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 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 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 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 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 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.000 0. Pearson.000 0. 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 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 1 7 49,655,000 51,000,000 0 Higher Order 2 4 37,868,000 36,450,000 2 Effects a 3 1 1,476,224 1,473,225 4 K-way Effects b 1 3 11,787,008 14,550,002 0 2 3 36,393,000 34,977,000 0 3 1 1,476,224 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 7. - 7.1 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 42 45 58 59 59 61 64 28 27 32 34 35 35 36 65 65 68 70 70 72 75 37 38 40 40 40 41 45 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 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 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 = 4.42 + 0.511x H υπ απ α a α 2.215 α b α 0.035, α υ α = 4.42 2.215 α b = 0.511 0.035. υ α α α α α α α α, a b, α α α. π Sig. α α απ 0.05. α α α α π α α α. α α π α πα υ υ α 7.1 α π α Include constant

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 = 29,97 30 4,42 + 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 ) 0 1025 6 1160 1 1055 12 1170 1.5 1060 15 1185 2 1065 20 1200 2.3 1070 35 1310 5 1060 55 1360 5.5 1095 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 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) 435,137 108,157 4,023,002 υ α π y = a + bx + cx 2 π α α Coefficients α : α = (Constant) = 435.137, b = toothsize = -0.859 α c = toothsize**2 = 0.000. ( SPSS ** α α ).

101 π π α π υ. toothsize**2 = 0.000 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 0,0004246. 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*1150 + 0,0004246*1150 2 = 9,26 435,137 Ν 0,858616*1250 + 0,0004246*1250 2 = 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*1250 + 0,0004246*1250 2 = 24,83 α υ 0 α α 0,0004246 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 1 1 + a 2 2 + a 3 3 + a 4 4

102 α α 7.5. α π α x υ π υ α α απ αφ απ α 1, 2, 3, 4. y 1 2 3 4 30 28 18 29 15 34 28 18 30 16 33 26 18 28 17 26 27 19 28 18 41 28 20 31 20 10 23 18 25 19 12 22 18 25 20 20 23 19 28 20 31 28 20 31 21 38 30 22 32 24 43 31 22 32 24 47 32 23 34 24 45 31 22 34 23 45 31 22 33 21 22 27 20 30 20 5 15 20 28 20 30 28 15 30 18 29 28 21 30 20 23 25 21 31 21 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 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 = -75.494 + 1.933 1 + 1.776 3 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. -75.412 12.218-6.172.000 1.882.386.621 4.876.000.212.959.035.221.829 1.990.774.418 2.572.022 -.465.659 -.098 -.705.492

104 Model 1 2 3 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. -75.412 12.218-6.172.000 1.882.386.621 4.876.000.212.959.035.221.829 1.990.774.418 2.572.022 -.465.659 -.098 -.705.492-75.037 11.710-6.408.000 1.870.370.617 5.058.000 2.063.677.433 3.047.008 -.367.473 -.077 -.776.450-75.494 11.549-6.537.000 1.933.356.638 5.425.000 1.776.560.373 3.172.006 7.2 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. 7.2.1 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 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 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 50 48 body mass Pearson Correlation,863 ** 1 Sig. (2-tailed),000 N 48 48 **. 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 50 48 body mass Correlation Coefficient,878 ** 1,000 Sig. (2-tailed),000. N 48 48 **. Correlation is significant at the 0.01 level (2-tailed). 7.2.2. (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 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 25 20 y = 0.679x - 1.6769 R² = 0.6302 15 floor 10 5 0 5 10 15 20 25 30 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 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 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 7.10. 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 0 8 8 floor Correlation,758 1,000,929 Significance (2-tailed),011,000 df 8 0 8 area Correlation,809,929 1,000 Significance (2-tailed),005,000 df 8 8 0 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 8. 8.1 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. 8.2 - 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 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 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 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. 8.3 - 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 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