Statistical Methods for the Biological, Environmental, and Health Sciences
Statistical Methods for the Biological, Environmental, and Health Sciences AMS 7
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This 8 page Class Notes was uploaded by Milton Sawayn DVM on Monday September 7, 2015. The Class Notes belongs to AMS 7 at University of California - Santa Cruz taught by David Draper in Fall. Since its upload, it has received 27 views. For similar materials see /class/182140/ams-7-university-of-california-santa-cruz in Applied Math And Statistics at University of California - Santa Cruz.
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Date Created: 09/07/15
AMS 7 Statistical Methods For the Biological and Environmental Sciences 2 Experimental Design David Draper Department of Applied Mathematics and Statistics University of California Santa Cruz draper amsucscedu http www ams ucsc eduNdraper 2006 David Draper all rights reserved Outline The ideas behind randomized controlled ex periments Completely randomized desig ns Blocking or matching to increase accuracy repeated measures designs Observational studies bias arising from po tential confounding factors and how to con trol for it 21 Randomized Controlled Experiments Case study psychobiology Does the psychological environment affect the anatomy of the brain in mammals This question was studied by some researchers at UC Berkeley about 40 years ago I notice that I39m uncertain about the extent to which environmental factors can affect brain anatomy so I decide to gather some data to reduce my uncertainty Broadly speaking there are two types of data gathering activities sample surveys drawing samples from populations and using the sampled data to estimate some aspect of the unsampled data and experiments settings in which people are interested in the effects of one or more treatment or independent variables on one or more outcome or dependent or response variables A sample survey wouldn t help much to answer the question above let39s see if we can think through how you might design an experiment to gain some insight Evidently from the framing of the question the independent variable we could call it X here is psychological environment and the dependent variable it could be called Y is brain anatomy we39re wondering if changes in the former can cause changes in the latter How should X and Y be measured here It s possible to envision a continuous range of psychological environments from enriched to deprived for simplicity let39s just imagine that it takes on only two values enriched versus deprived so X is dichotomous Sample Size Determination It s not ethical to experiment on something like this with humans the researchers at Berkeley worked with laboratory animals rats and to measure Y they decided to sacrifice the rats and measure the weight of the cortex the gray matter or so called thinking part of the brain in each animal in mg To make operational the idea of enriched and deprived psychological environments it turns out that rats are social and playful animals similar to humans in some ways so the Berkeley researchers decided to allow the enriched environment rats to live 12 to a cage and to provide them with toys that were changed daily by contrast the deprived environment rats lived alone with no toys Here39s a bad design to get the discussion going Get a bunch of rats subject all of them to an enriched psychological environment wait awhile see what cortex weights result Even this design needs further elaboration How many is a bunch How long is awhile Answering the question How many is an example of sample size determination something that we39ll talk about in some detail later for now it39s enough to notice that a it39s certainly possible to have too little data suppose you ran the experiment with only say 2 rats this would leave you with too much uncertainty to be useful and b possibly a bit surprisingly it39s also possible to have too much data it turns out that in most problems beyond a certain point there39s little value in decreasing your uncertainty any farther and in a world where we39re always making choices in the face of time and money constraints effort made to decrease uncertainty past this point would be better spent on working on a different problem 4 Factual and Counterfactual Data In the Berkeley brain anatomy experiment the researchers worked with n 120 rats after making a sample size calculation like those we ll talk about later and this turned out to be a good choice As for the length of the experiment the researchers were interested in the effect of early life psychological environment on brain anatomy so they waited until all of the animals had reached adult maturity before sacrificing them One way to visualize the raw data arising from design 0 would be as follows Enriched Rat Cortex Number Weight 1 y1 689 w 656 n 120 120 yn 649 Mean g 683 mg SD 8 32 mg OK the enriched rats had a mean cortex weight of 683 mg this is data about what actually happened under design 0 does this show that enrichment has caused an increase in mean cortex weight Well no because we have no point of comparison we don39t know what the mean cortex weight of these same rats W if they had received the deprived environment instead this is data about what would have happened if all of the rats had been raised in the deprived psychological environment instead Causal Inference Another way to visualize the raw data from design 0 is the basis of what s called the counterfactual model of causal inference for each rat we imagine what the cortex weight would have been if it had received the enriched environment and what the cortex weight would have been if instead the same rat had received the deprived environment the factual data is filled in as numbers and the counterfactual data as question marks Rat Cortex Weight If Number Enriched Deprived Design 0 1 689 7 66 n 120 120 649 Mean Q1 2 683 mg SD 81 32 mg If you could observe both the enriched and deprived columns in this table you39d be able to estimate the causal effect of the enriched versus deprived environment on cortex weight for any given rat you could calculate enriched cortex weight deprived cortex weight and that would be the causal effect for that animal you could then calculate the mean of these differences as a summary of the typical causal effect Because the data values in the deprived environment column of this table are entirely missing with design 0 we have no way with this design to estimate the causal effects evidently we need a design in which some rats get the enriched environment and some get the deprived environment Let39s call the rats in the enriched environment the treatment group T and the rats in the deprived environment the control group C Treatment and Control Groups A design with T and C groups in which the investigators have control over which subjects in this case rats go into which groups is called a controlled experiment How many rats should we put in each group The basic principle turns out to be intuitively reasonable you should concentrate your replications in other words gather the most data where your uncertainty is largest sometimes you39ll have more uncertainty about the outcome variable values under the T condition than the C and in those situations you should put more of the experimental subjects rats in this case into the T group but here we39re equally uncertain about both T and C and it39s both natural and optimal to put half of the rats in each group All of this motivates Get 120 lab rats put 60 in the T group and 60 in C wait til maturity and measure their cortex weights The causal inference data set for design 1 would look like this Rat T or C Cortex Weight If Number Status Enriched Deprived Design 1 1 T 689 60 T 649 61 C 657 l o 2 7 602 Mean 371 683 mg 372 647 mg SD 81 32 mg 8230 mg The means and SDs here are computed ignoring the missing data the question marks OK so now with design 1 we can say that the mean cortex weight of the enriched animals was 371 372 683 647 36 mg larger than the mean cortex weight of 7 Potential confounding Factors the deprived rats and W t 1056 so the cortex weights of the T rats were on average 56 larger than the C animals this difference would be judged by researchers in psychobiology as large in practical terms but can we conclude that this difference was caused by the psychological environment Suppose that some genetic strains of rats tend to have smaller cortexes than others and suppose further that without meaning to it turned out that I put most of the genetically small cortex rats into the control group this would be sufficient to explain the results in the data set on the previous page even if the psychological environment had no effect Let39s agree to call two variables if there39s a tendency for one of them to go up or down on average as the other one changes Then according to this definition there might be an association between the outcome variable Y cortex weight in this experiment and a third variable Z genetic background because as you move from one value of Z to another in other words from one genetic background to another Y cortex weight goes up or down on average Furthermore if without meaning to I put most of the genetically small cortex rats into the C group then there would also be an association between Z and the treatment variable X which here is the dichotomy T versus C since in that case as you move from the treatment to the control group the genetic backgrounds of the rats would change on average In an experimental setting with a treatment variable X and an outcome variable Y any third variable Z that may plausibly be associated both with X and with Y is called a potential confounding factor PCF 8
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