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Linear Models

Linear Models. Two-Way ANOVA. Example -- Background. Bacteria -- effect of temperature (10 o C & 15 o C) and relative humidity (20%, 40%, 60%, 80%) on growth rate (cells/d ). 120 petri dishes with a growth medium available

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Linear Models

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  1. Linear Models Two-Way ANOVA

  2. Example -- Background • Bacteria -- effect of temperature (10oC & 15oC) and relative humidity (20%, 40%, 60%, 80%) on growth rate (cells/d). • 120 petri dishes with a growth medium available • Growth chambers where all environmental variables can be controlled. • What is the response variable, factor(s), level(s), treatment(s), replicates per treatment? LM ANOVA 2

  3. Factorial or Crossed Design • Each treatment is a combination of both factors. LM ANOVA 2

  4. Factorial or Crossed Design • Advantages (over two OFAT experiments) • Efficiency – each individual “gives information” about each level of BOTH factors. OFAT LM ANOVA 2

  5. Factorial or Crossed Design • Advantages (over two OFAT experiments) • Efficiency – each individual “gives information” about each level of BOTH factors. • Power – increased due to increased effective n. • Effect Size – detect smaller differences • Interaction effect– can be detected. LM ANOVA 2

  6. Interaction Effect • Effect of one factor on the response variable differs depending on level of the other factor. LM ANOVA 2

  7. No Interaction Effect LM ANOVA 2

  8. Main Effects • Differences in “level” means for a factor • “Strong” relative humidity main effect • “Weak” temperature main effect. 11.25 10.25 6.5 9.5 12.5 14.5 LM ANOVA 2

  9. Main Effects • “Strong” relative humidity main effect • “Weak” temperature main effect. LM ANOVA 2

  10. Interaction Effect LM ANOVA 2

  11. Humidity and Temperature Effects LM ANOVA 2

  12. Humidity Effect Only LM ANOVA 2

  13. Temperature Effect Only LM ANOVA 2

  14. No Effects LM ANOVA 2

  15. Example #1 × Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ √ LM ANOVA 2

  16. Example #2 × Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ × LM ANOVA 2

  17. Example #3 Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ LM ANOVA 2

  18. Example #4 × Interaction Effect Factor 1 Main Effect Factor 2 Main Effect × √ LM ANOVA 2

  19. Example #5 Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ LM ANOVA 2

  20. Example #6 Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ LM ANOVA 2

  21. Example #7 × Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ √ LM ANOVA 2

  22. Example #8 Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ LM ANOVA 2

  23. Example #9 × Interaction Effect Factor 1 Main Effect Factor 2 Main Effect × √ LM ANOVA 2

  24. Example #10 Interaction Effect Factor 1 Main Effect Factor 2 Main Effect √ LM ANOVA 2

  25. Terminology / Symbols • One factor is “row” factor • r = number of levels • Other factor is “column” factor • c = number of levels • Yijk= response variable for kth individual in ithlevel of row factor and jth level of column factor • for simplicity, assume n is same for all i,j LM ANOVA 2

  26. Terminology / Symbols Grand mean Level means Treatment means `Y11. `Y1.. `Y12. `Y1c. `Y21. `Y22. `Y2c. `Y2.. `Yr1. `Yr2. `Yrc. `Yr.. `Y.1. `Y.2. `Y.c. `Y... LM ANOVA 2

  27. 2-Way ANOVA Purpose • Determine significance of interaction and, if appropriate, two main effects. • Are differences in means “different enough” given sampling variability? LM ANOVA 2

  28. 2-Way ANOVA Calculations • MSWithin is variability about ultimate full model • MSTotal is variability about ultimate simple model • if MSAmong is large relative to MSWithin then ultimate full model is warranted • i.e., some difference in treatment means • implies differences due to row factor, column factor, or interaction between the two • SSAmong= SSRow+ SSCol + SSInteraction • If MSRow is large relative to MSWithin then a difference due to the row factor is indicated • Similar argument for column and interaction effects LM ANOVA 2

  29. 2-Way ANOVA Calculations SSAmong = SSRow + SSColumn + SSInteraction LM ANOVA 2

  30. r ( ) 2 å SSRow=cn - Y Y .. ... i = i 1 2-Way ANOVA Calculations c ( ) 2 å SSColumn=rn - Y Y . . ... j = i 1 `Y11. `Y1.. `Y12. `Y1c. `Y21. `Y22. `Y2c. `Y2.. `Yr1. `Yr2. `Yrc. `Yr.. `Y.1. `Y.2. `Y.c. `Y... LM ANOVA 2

  31. Two-Way ANOVA Table Source df SS MS F . Row r-1 SSRowSSRow/[r-1] MSRow/MSWithin Column c-1 SSColSSCol/[c-1] MSCol/MSWithin Inter (r-1)(c-1) SSIntSSInt/[(r-1)(c-1)] MSInt/MSWithin Within rc(n-1) SSWithinSSWithin/[rc(n-1)] Total rcn-1 SSTotal LM ANOVA 2

  32. Example • What is the optimal temperature (27,35,43oC) and concentration (0.6,0.8,1.0,1.2,1.4% by weight) of the nutrient, tryptone, for culturing the Staphylococcus aureus bacterium. Each treatment was repeated twice. The number of bacteria was recorded in millions CFU/mL (CFU=Colony Forming Units). LM ANOVA 2

  33. Review Handout – Example 1 • lm() • anova() • glht() • fitPlot() • addSigLetters() LM ANOVA 2

  34. Assumptions and Checking in R • Same as for the one-way ANOVA LM ANOVA 2

  35. Example • Measured soil phosphorous levels in plots near Sydney, Australia. • Each plot was characterized by type of soil (shale- or sandstone-derived) and “topographic” location (valley, north, south, or hillside). • Data in SoilPhosphorous.txt • Does mean soil phosphorous level differ by soil type or topographic location? • Is there an interaction effect? LM ANOVA 2

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