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# BTM8106 Week 4 Complete Solution

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Date Created: 11/05/15

Jackson evennumbered Chapter Exercises pp 335337 Experimental Designs 2 Explain the difference between multiple independent variables and multiple levels of independent variables Which is better Answer The general purpose of multivariate analysis of variance MANOVA is to determine whether multiple levels of independent variables on their own or in combination with one another have an effect on the dependent variables MANOVA requires that the dependent variables meet parametric requirements MANOVA is used under the same circumstances as ANOVA but when there are multiple dependent variables as well as independent variables within the model which the researcher wishes to test MANOVA is also considered a valid alternative to the repeated measures ANOVA when sphericity is violated Like an ANOVA MANOVA examines the degree of variance within the independent variables and determines whether it is smaller than the degree of variance between the independent variables If the within subjects variance is smaller than the betw een subjects variance it means the independent variable has had a significant effect on the dependent variables There are two main differences between MANOVAs and ANOVAs The first is that MANOVAs are able to take into account multiple independent and multiple dependent variables within the same model permitting greater complexity Secondlyrather than using the F value as the indicator of significance a number of multivariate measures MANOVAs the independent variables relevant to each main effect are weighted to give them priority in the calculations performed In interactions the independent variables are equally weighted to determine whether or not they have an additive effect in terms of the combined variance they account for in the dependent variables The main effects of the independent variables and of the interactions are examined with all else held constant The effect of each of the independent variables is tested separately Any multiple interactions are tested separately from one another and from any significant main effects Assuming there are equal sample sizes both in the main effects and the inter actions each test performed will be independent of the next or previous calculation exce pt for the error term which is calculated across the independent variables 3 What is blocking and how does it reduce quotnoisequot What is a disadvantage of blocking Sol The Randomized Block Design is research design39s equivalent to stratified random sampling Like stratified sampling randomized block designs are constructed to reduce noise or variance in the data see Classifying the Exper imental Designs How do they do it They require that the researcher divide the sample into relatively homogeneous subgroups or blocks analogous to quotstrataquot in stratified sampling Then the experimental design you want to impl ement is implemented within each block or homogeneous subgroup The key idea is that the variability within each block is less than the variability of the entire sample Thus each estimate of the treatment effect within a block is more efficient than estimates across the entire sample And when we pool these more efficient estimates across blocks we should get an overall more efficient estimate than we would without blocking How Blocking Reduces Noise So how does blocking work to reduce noise in the data To see how it works you have to begin by thinking about the nonblocked study The figure shows the pretestposttest distribution for a hypothetical prepost randomized experi mental design We use the 39X39 symbol to indicate a program group case and the 39O39 symbol for a comparison group member You can see that for any specific pretest value the program group tends to outscore the comparison group by about 10 points on the posttest That is there is about a 10point posttest mean difference a a an a a m EIII pretest Now let39s consider an example where we divide the sample into three relatively homogeneous blocks To see what happens graphically we39ll use the pretest mea sure to block This will assure that the groups are very homogeneous Let39s look at what is happening within the third block Notice that the mean difference is still the same as it was for the entire sample about 10 points within each block But also notice that the variability of the posttest is much less than it was for the entire samp le Remember that the treatment effect estimate is a signaltonoise ratio The signal in this case is the mean difference The noise is the variability The two figures show that we haven39t changed the signal in moving to blocking there is still about a 10 point posttest difference But we have changed the noise the variability on the posttest is much smaller within each block that it is for the entire sample So the treatment effect will have less noise for the same signal pus est 100 ED ED TD ED ED 40 ED 2 40 an EU D ED pre est It should be clear from the graphs that the blocking design in this case will yield the stronger treatment effect But this is true only because we did a good job assuring that the blocks were homogeneous If the blocks weren39t homogeneous their variability was as large as the entire sample39s we would actually get worse estimates than in the simple randomized experim ental case 4 What is a factor How can the use of factors bene t a design Sol Probably the easiest way to begin understanding factorial designs is by looking at an example Let39s imagine a design where we have an educational program where we would like to look at a variety of program variations to see which works best For instance we would like to vary the amount of time the children receive instruction with one group getting 1 hour of instruction per week and another getting 4 hours per week And we39d like to vary the setting with one group getting the instruction inclass probably pulled off into a comer of the classroom and the other group being pulledout of the classroom for instruction in another room We could think about having four separate groups to do this e In Instructinn 1 huurfweek l l4 hoursnnreek l F ulluut lnclass With factorial designs we don39t have to compromise when answering these questions We can have it both ways if we cross each of our two time in inst ruction conditions with each of our two settings Let39s begin by doing some defining of terms In factorial designs a factor is a major independent variable In this example we have two factors time in instruction and setting A level is a subdivision of a factor In this example time in instruction has two levels and setting has two levels Sometimes we depict a factorial design with a numbering notation In this example we can say that we have a 2X 2 spoken quottwobytwo factorial design In this notation the number of numbers tells you how many factors there are and the number values tell you how many levels If I said I had a 3 X 4 factorial design you would know that I had 2 factors and that one factor had 3 levels while the other had 4 Order of the numbers makes no difference and we could just as easily term this a 4 X 3 factorial design The number of different treatment groups that we have in any factorial design can easily be deter mined by multiplying through the number notation For instance in our example we have 2X 2 4 groups In our notational example we would need 3 X 4 12 groups 5Expain main effects and interaction effects Sol The Main Effects A main effect is an outcome that is a consistent difference between levels of a factor For instance we would say there39s a main effect for setting if we find a statistical difference between the averages for the inclass and pullout groups at all levels of time in instruction The rst figure depicts a main effect of time For all settings the 4 hourweek condition worked better than the l hourweek one It is also possible to have a main effect for setting and none for time Main Effects Time 1 hr dl39lra out E E in 391hr 4hrs E 39uut in In the second main effect graph we see that in class training was better than pull mmhmmq mwhmmq 1hr diva I III out training for all amounts of time Main Effects Time 1 hr dl39lra Finally it is possible to have a main effect on both variables simultaneously as depicted in the third main effect figure In this instance 4 hoursweek always works better than 1 hourweek and inclass setting always works better than pull out Main Effects Mh 39 IIIIII out in 1 hr ahrsa Interaction Effects If we could only look at main effects factorial designs would be useful But because of the way we combine levels in factorial designs they also enable us to examine the interaction effects that eXist between factors An interaction effect eXists when differences on one factor depend on the level you are on another factor It39s important to recognize that an interaction is between factors not levels We wouldn39t say there39s an interaction between 4 hoursweek and in class treatment Instead we would say that there39s an interaction between time and setting and then we would go on to describe the specific levels involved Interactien Effects Time 1 hr dl39lre out til I E in 391hr 4 01 4hre i E mwhmmq rowelmead 1hr d11re 4 I D How do you know if there is an interaction in a factorial design There are three ways you can determine there39s an interaction First when you run the statistical analysis the statistical table will report on all main effects and interactions Second you know there39s an interaction when can39t talk about effect on one factor without mentioning the other factor if you can say at the end of our study that time in instruction makes a difference then you know that you have a main effect and not an interaction because you did not have to mention the setting factor when describing the results for time On the other hand when you have an interaction it is impossible to describe your results accurately without mentioning both factors Finally you can always spot an interaction in the graphs of group means whenever there are lines that are not parallel there is an interaction present If you check out the main effect graphs above you will notice that all of the lines within a graph are parallel In contrast for all of the interaction graphs you will see that the lines are not parallel Interaction Effects Time 1 hr dl39lra out in X 391hr 4hr5 Settlng l X 3 much31034 minimum314 out in 1l1r ahra 6How does a covariate reduce noise Sol One of the most important ideas in social research is how we make a statistical adjustment adjust one variable based on its covariance with another variable If you understand this idea you39ll be well on your way to mastering social research What I want to do here is to show you a series of graphs that illustrate pictorially what we mean by adjusting for a covariate Let39s begin with data from a simple ANCOVA design as described above The rst gure shows the prepost bivariate distribution Each quotdotquot on the graph represents the pretest and posttest score for an individual We use an 39X39 to signify a program or treated case and an 39O39 to describe a control or comparison case You should be able to see a few things immediately First you should be able to see a whopping treatment effect It39s so obvious that you don39t even need statistical analysis to tell you whether there39s an effect altho ugh you may want to use statistics to estimate its size and probability How do I know there39s an effect Look at any pretest value value on the horizontal axis Now look up from that value you are looking up the posttest scale from lower to higher posttest scores Do you see any pattern with respect to the groups It should be obvious to you that the program cases the 39X39s tend to score higher on the posttest at any given pretest value Second you should see that the posttest var iability has a range of about 70 points Now let39s fit some straight lines to the data The lines on the graph are regression lines that describe the prepost relationship for each of the groups The regression line shows the expected posttest score for any pretest score The treatment effect is even clearer With the regression lines You should see that the line for the treated group is about 10 points higher than the line for the comparison group at any pretest value 7Describe and explain three tradeoffs present in experiments Sol 1 People Make Tradeoffs Economic goods and services are limited while the need to use services of these goods and services seem limitless There are simply not enough goods and services to satisfy even a small fraction of everyone39s cons umption desires Thus societies must decide how to use these limited resources and distribute them among different people This means to get one thing that we like we usually have to give up another thing that we also like Making decision requires trading off one goal against another Consider a society that decides to spend more on national defense to protect its shores from foreign aggressors the more the society spends on the national defense the less it can spend on personal goods to raise its standard of living at home Or consider the tradeoff between a clean air environment and a high level of income Laws that require firms to reduce pollution have the cost of reducing the incomes of the firm39s owners workers and customers while pollution regulations give the society the benefit of a cleaner environment and the improved health that comes with it Another trade off society faces is between efficiency and equity Efficiency deals with a society39s ability to get the most effective use 0 its resources in satisfying people39s wants and needs Equity denotes the fair distribution of benefits of those resources among society39s members

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