Nested designs Applied Statistics and Experimental Design Chapter 7 - PowerPoint PPT Presentation

Split plot designs Mixed effects modeling Nested designs Applied Statistics and Experimental Design Chapter 7 Peter Hoff Statistics, Biostatistics and the CSSS University of Washington

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Potato example: Sulfur added to soil kills bacteria, but too much sulfur can damage crops. Researchers are interested in comparing two levels of sulfur additive (low, high) on the damage to two types of potatoes. Factors of interest: 1. Potato type ∈ { A , B } 2. Sulfur additive ∈ { low,high } Experimental material: Four plots of land. Design constraints: • It is easy to plant different potato types within the same plot • It is difficult to have different sulfur treatments in the same plot, due to leeching.

Split plot designs Mixed effects modeling Split plot design Split-plot design 1. Each sulfur additive was randomly assigned to two of the four plots. 2. Each plot was split into four subplots. Each potato type was randomly assigned to two subplots per plot. A B B A L H A B A B B A B A H L A B A B

Split plot designs Mixed effects modeling Split plot design Split-plot design 1. Each sulfur additive was randomly assigned to two of the four plots. 2. Each plot was split into four subplots. Each potato type was randomly assigned to two subplots per plot. A B B A L H A B A B B A B A H L A B A B

Split plot designs Mixed effects modeling Split plot design Split-plot design 1. Each sulfur additive was randomly assigned to two of the four plots. 2. Each plot was split into four subplots. Each potato type was randomly assigned to two subplots per plot. A B B A L H A B A B B A B A H L A B A B

Split plot designs Mixed effects modeling Randomization Sulfur type was randomized to whole plots; Potato type was randomized to subplots. Initial data analysis: Sixteen responses, 4 treatment combinations. • 8 responses for each potato type • 8 responses for each sulfur type • 4 responses for each potato × type combination > f i t . f u l l < − lm ( y˜ type ∗ s u l f u r ) ; f i t . add < − lm ( y˜ type+s u l f u r ) > anova ( f i t . f u l l ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 13.4459 0.003225 ∗∗ s u l f u r 1 0.54022 0.54022 4.8803 0.047354 ∗ type : s u l f u r 1 0.00360 0.00360 0.0325 0.859897 R e s i d u a l s 12 1.32835 0.11070 > anova ( f i t . add ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 14.5270 0.00216 ∗∗ s u l f u r 1 0.54022 0.54022 5.2727 0.03893 ∗ R e s i d u a l s 13 1.33195 0.10246

Split plot designs Mixed effects modeling Randomization Sulfur type was randomized to whole plots; Potato type was randomized to subplots. Initial data analysis: Sixteen responses, 4 treatment combinations. • 8 responses for each potato type • 8 responses for each sulfur type • 4 responses for each potato × type combination > f i t . f u l l < − lm ( y˜ type ∗ s u l f u r ) ; f i t . add < − lm ( y˜ type+s u l f u r ) > anova ( f i t . f u l l ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 13.4459 0.003225 ∗∗ s u l f u r 1 0.54022 0.54022 4.8803 0.047354 ∗ type : s u l f u r 1 0.00360 0.00360 0.0325 0.859897 R e s i d u a l s 12 1.32835 0.11070 > anova ( f i t . add ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 14.5270 0.00216 ∗∗ s u l f u r 1 0.54022 0.54022 5.2727 0.03893 ∗ R e s i d u a l s 13 1.33195 0.10246

Split plot designs Mixed effects modeling Randomization Sulfur type was randomized to whole plots; Potato type was randomized to subplots. Initial data analysis: Sixteen responses, 4 treatment combinations. • 8 responses for each potato type • 8 responses for each sulfur type • 4 responses for each potato × type combination > f i t . f u l l < − lm ( y˜ type ∗ s u l f u r ) ; f i t . add < − lm ( y˜ type+s u l f u r ) > anova ( f i t . f u l l ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 13.4459 0.003225 ∗∗ s u l f u r 1 0.54022 0.54022 4.8803 0.047354 ∗ type : s u l f u r 1 0.00360 0.00360 0.0325 0.859897 R e s i d u a l s 12 1.32835 0.11070 > anova ( f i t . add ) Df Sum Sq Mean Sq F v a l u e Pr( > F) type 1 1.48840 1.48840 14.5270 0.00216 ∗∗ s u l f u r 1 0.54022 0.54022 5.2727 0.03893 ∗ R e s i d u a l s 13 1.33195 0.10246