Operational Trials: Data Analysis Wendy Bergerud Research Branch - PowerPoint PPT Presentation

Operational Trials: Data Analysis Wendy Bergerud Research Branch BC Min. of Forests May 2003

Proposed Analysis � When designing the trial we should consider what form of analysis we expect to run on the data. � What design requirements will this proposed analysis have? � Can these design requirements be met? Q12 WAB

Proposed Analysis � Should additional variables be collected and/or are some unnecessary? � Does the analysis require some assumptions that the design can’t meet? � Are some treatment combinations necessary or not? � Should all treatment combinations be equally replicated? WAB

Types of Variables � Measured or continuous variables � Counts � Categorical variables � Non-ordered � Ordered � Dichotomous (proportions) � Response vs. Explanatory Variables WAB

Response Variables � What variables measure the response we are interested in and how will we measure them? � Can the variables we are interested in be directly measured or must we use a proxy or surrogate measure? Q10 WAB

What is a model? � A model is simply a description that summarizes the groups and/or relationships that we think exist in the subject matter that we are studying. � Statistical models often have a formal mathematical representation. � But these models are developed from the ideas that we have about the subject matter we are studying. WAB

Statistical Models for Data Analysis Type of Explanatory General Linear Variable Model (GLM) Categorical: 2 levels t -test 2 or more levels ANOVA & contrasts Continuous: 1 variable Simple Regression 2 or more variables Multiple Regression Categorical & Continuous: ANCOVA 2 or more & 1 or more � These are for continuous response variables, modelled using the normal distribution. WAB

Other Models for Data Analysis � The previous models can be adapted for other types of response variables: � Counts Poisson distribution � Proportions binomial distribution � Contingency tables can also be used to test counts and proportions obtained from simple study designs. WAB

Data Analysis Steps 1 Look for patterns, errors, and outliers. 2 Correct data as necessary. 3 Fit models and test terms in the model. 4 Examine selected model for adequate fit. � Repeat any steps as necessary. 5 Summarize data analysis and the final fitted model. WAB

Data Analysis Example DBH (cm) Height (m) BWBS 23.1 26.6 19.3 4.1 26.1 35.1 24.0 24.6 22.3 27.2 19.9 21.5 26.6 25.9 24.0 24.7 20.5 22.4 20.2 20.4 52.6 . 27.5 . 22.2 . 22.7 . SBS 35.0 34.2 31.2 25.5 21.1 22.3 22.4 19.3 24.2 27.5 27.2 24.1 34.3 26.6 27.2 24.9 26.3 . 28.2 . 26.8 . 26.5 . 26.9 . 27.6 . D1 WAB

Step 1 -- Stem and Leaf Plots BWBS SBS Diameter S Stem tem Le Leaf af # # Stem Ste m Leaf Leaf # # 34| 34|23 230 0 3 3 5| 5|3 3 1 1 32| 32| 4| 4| 30| 30| 4| 4| 28| 28| 3| 3|5 5 1 1 26| 26|36 36895 895 5 5 3| 3| 24| 24|2 2 1 1 2| 2|6677 66777 7 5 5 22| 22|3 3 1 1 2| 2|0222 02223 3 5 5 20| 20|1 1 1 1 | |---- ----+---- +----+-- +----+ --+----+ ----+ | |----+-- --+----+-- --+----+--- -+----+ -+ Mu Mult ltiply iply Stem Stem & & Lea Leaf by f by 10 10 � One unusually high diameter in BWBS. � Two groups in SBS. D2 WAB

Step 1 -- Stem and Leaf Plots BWBS SBS Diameter S Stem tem Le Leaf af # # Stem Ste m Leaf Leaf # # 34| 34|23 230 0 3 3 5| 5|3 3 1 1 32| 32| 4| 4| 30| 30| 4| 4| 28| 28| 3| 3|5 5 1 1 26| 26|36 36895 895 5 5 3| 3| 24| 24|2 2 1 1 2| 2|6677 66777 7 5 5 22| 22|3 3 1 1 2| 2|0222 02223 3 5 5 20| 20|1 1 1 1 | |---- ----+---- +----+-- +----+ --+----+ ----+ | |----+-- --+----+-- --+----+--- -+----+ -+ Mu Mult ltiply iply Stem Stem & & Lea Leaf by f by 10 10 � One unusually high diameter in BWBS. � Two groups in SBS. D2 WAB

Step 1 -- Stem and Leaf Plots BWBS SBS Height Stem Stem Leaf Leaf # # Stem Leaf Stem Leaf # 30|2 30 |2 1 1 2| 2|558 558 3 28 28|2 |2 1 1 2| 2|0002344 0002344 7 26|5226 26 |5226 4 4 1| 1|9 9 1 24|195 24 |195 3 3 1| 1| 22|4 22 |4 1 1 0| 0| 20| 20 0|4 0| 4 1 18 18|3 |3 1 1 | |----+--- ---+----+-- -+----+- -+----+ ---+ |----+----+- |----+----+----+----+ ---+----+ Multiply Ste Mult iply Stem & & Leaf Leaf by 10 by 10 � Two unusually low heights, one in each zone. � The SBS value is actually 19.3. D3 WAB

Step 1 -- Stem and Leaf Plots BWBS SBS Height Stem Leaf Stem Leaf # # Stem Stem Leaf Leaf # 30|2 30 |2 1 1 2|558 2| 558 3 28|2 28 |2 1 1 2|0002344 2| 0002344 7 26|5226 26 |5226 4 4 1| 1|9 9 1 24|195 24 |195 3 3 1| 1| 22 22|4 |4 1 1 0| 0| 20 20| 0|4 0| 4 1 18|3 18 |3 1 1 |----+--- | ---+----+-- -+----+- -+----+ ---+ |----+----+- |----+----+----+----+ ---+----+ Mult Multiply Ste iply Stem & & Leaf Leaf by 10 by 10 � Two unusually low heights, one in each zone. � The SBS value is actually 19.3. D3 WAB

Step 1 -- Boxplots 60 35 30 50 25 D i H 20 a e m i 40 e g t h 15 e t r 10 30 5 20 0 BWBS SBS BWBS SBS BEC-Zone BEC-Zone D4 WAB

Step 1 -- Scatterplot 40 30 Height 20 10 0 20 30 40 50 60 Diameter BEC-Zone BWBS SBS D5 WAB

Data Analysis Example DBH (cm) Height (m) BWBS 23.1 26.6 19.3 4.1 26.1 35.1 24.0 24.6 22.3 27.2 19.9 21.5 26.6 25.9 24.0 24.7 20.5 22.4 20.2 20.4 52.6 . 27.5 . 22.2 . 22.7 . SBS 35.0 34.2 31.2 25.5 21.1 22.3 22.4 19.3 24.2 27.5 27.2 24.1 34.3 26.6 27.2 24.9 26.3 . 28.2 . 26.8 . 26.5 . 26.9 . 27.6 . D1 WAB

Step 2 -- Correct data � We have found three unusual points: � Are these ‘real’ values? If not, what should the values be? � Sometimes the unusual data points have the most interesting story to tell - don’t casually throw them out! � Suppose that the height of 4.1 should be 24.7, but that the other values are okay. D6 WAB

Step 3 -- Fit a model � Separate response variables: � Analyse dbh and height separately with a t -test or ANOVA. � Are the two groups different? � Relationship between response variables: � Analyse the relationship between height and diameter. � Is the relationship the same for both groups? D7 WAB

Step 3 -- Consider Five models Model Name Description One group 1 Data belong to just one group. 2 Two groups Data belong to two groups defined by BEC-Zone. 3 One line There is a linear relationship between height and diameter but it is the same for the two groups. 4 Two parallel There is a linear relationship between height and diameter but the line for one group is higher than for lines the other group, while both have the same slope. Two lines 5 There is a linear relationship between height and diameter but the slope for one group is steeper than for the other group. D8 WAB

Step 3 -- Parallel Line Model 35 35 30 30 Height 25 25 20 20 15 15 20 30 40 50 60 Diameter BEC-Zone BWBS SBS BEC-Zone BWBS SBS D9 WAB

Step 3 -- Model Fit Degrees of Residual Sums Residual Model Name Freedom of Squares Mean Square (df) (SSR) (MSR) One group 1 22 223.3565 10.15 2 Two groups 21 170.4710 8.12 3 One line 21 147.8133 7.04 4 Two parallel 20 96.7094 4.84 lines 5 Two lines 19 89.8038 4.73 D10 WAB

Step 3 -- Comparing the models Models Difference Testing: Used in SSR F-test (df) p-value Result Yes † 1. Are the lines parallel? 4 & 5 6.91 1.5 (1, 19) p = 0.24 Given that lines are parallel: 2. Can we use just one line? 3 & 4 51.10 10.5 (1, 20) p = 0.0040 No 3. Can we use just two groups? 2 & 4 73.76 15.25 (1,20) p = 0.0046 No † Technically we can’t accept a null hypothesis, but in order to proceed we must make decisions, even if they might be wrong. � Parallel Lines model provides a good overall fit. � Significance is easy to see when p-values are presented with just two significant digits. D11 WAB

Step 4 - Checking the model � We do this my examining the residuals: � Are they normally distributed? (or at least, symmetric?) � Do they have any patterns with respect to the explanatory variables and the fitted values? � Does their variability look similar regardless of the explanatory variables? D12 WAB

Step 4 - Checking the model S Stem Le Leaf af # Bo Boxp xplo lot 3 3 5 5 1 | | 2 2 02 0233 338 8 5 + +-----+ 1 1 35 356 6 3 | | | 0 0 9 9 1 | | + + | - -0 76 7632 32 4 * *-----* - -1 97 9775 7520 20 6 + +-----+ - -2 30 30 2 | | - -3 | | - -4 | | - -5 1 1 1 | | -- ---- --+- +----+----+- -+--- ---+ -+ � Stem & Leaf Plot with Boxplot (from SAS’s Proc Univariate ) D13 WAB

Step 4 - Checking the model S Stem Le Leaf af # Bo Boxp xplo lot 3 3 5 5 1 | | 2 2 02 0233 338 8 5 + +-----+ 1 1 35 356 6 3 | | | 0 0 9 9 1 | | + + | - -0 76 7632 32 4 * *-----* - -1 97 9775 7520 20 6 + +-----+ - -2 30 30 2 | | - -3 | | - -4 | | - -5 1 1 1 | | -- ---- --+- +----+----+- -+--- ---+ -+ � Stem & Leaf Plot with Boxplot (from SAS’s Proc Univariate ) D13 WAB