Week 2 Basic Statistical Concepts, Part II

Transcription

1 Week 2 Basic Statistical Concepts, Part II

2 Week 2 Objectives 1 Data presentation through numerical and graphical summaries using R: sample mean, variance and percentiles; the box plot, histogram, stem and leaf diagram, the pie chart and bar graph, the scatter plot and scatter plot matrix. 2 The basics of comparative studies including randomization, confounding and Simpson s paradox. 3 Statistical experiments vs observational studies, and their relevance for establishing causation. 4 Factorial designs concepts: main effects and interactions. 5 Use of R for comparative graphics, the interaction plot, and for computing the main effects and interactions.

3 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

4 Outline Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

5 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Mean, Variance and Standard Deviation With the data set in the R object x, use: mean(x) # for the mean var(x); sd(x) # for the variance and standard deviation If the population is categorical, table(x); table(x)/length(x) return the sizes and proportions of the categories, respectively. If v contains the statistical population use var(v)*(length(v)-1)/length(v) for the population variance.

6 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Example The productivity of each of the N = 10, 000 employees of a company is rated on a scale from 1-5. Let the statistical population v 1, v 2,..., v 10,000 be v i = 1, i = 1,..., 300, v i = 2, i = 301,..., 1, 000, v i = 3, i = 1, 001,..., 5, 000, v i = 4, i = 5, 001,..., 9, 000, v i = 5, i = 9, 001,..., 10, 000. Find the population proportions for each rating category, the average rating, and the population variance and standard deviation of rating.

7 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Solution. Set the statistical population in v: v=c(rep(1,300),rep(2,700),rep(3,4000),rep(4,4000),rep(5,1000)) Compute the proportions: table(v)/10000 Compute the mean, variance and standard deviation: mean(v); var(v)*(length(v)-1)/length(v) sqrt(var(v)*(length(v)-1)/length(v))

8 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Example Take a simple r.s. of size n = 500 from the population of employees from the previous example, and compute the sample proportions of the different ratings, the average rating, and the sample variance and standard deviation. Solution. x=sample(v, size = 500) table(x)/500 mean(x); var(x); sd(x)

9 Sample Percentiles Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots With the data set in the object x, the commands median(x) quantile(x,0.25) quantile(x,c(0.3,0.7,0.9)) summary(x) R commands for percentiles give, respectively, the median, the 25th percentile, the 30th, 70th and 90th percentiles, and a five number summary of the data consisting of x (1), q 1, x, q 3, and x (n). [summary(x) also gives the sample median.]

10 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Example Scientists have been monitoring the ozone hole since See the images shown in The 14 Ozone measurements (Dobson units) given in OzoneData set are taken in 2002 from the lower stratosphere, between 9 and 12 miles altitude. Obtain the five number summary as well as the 70th, 80th and 90th percentiles. Solution: Read the data in the R object oz using oz = read.table( http: //media.pearsoncmg.com/cmg/pmmg_mml_shared/mathstatsresources/akritas/ozonedata.txt, header =T) and use x=oz$ozonedata; summary(x); quantile(x, c(0.7, 0.8, 0.9)). NOTE: By typing the commands oz; x you can see the difference between the data frame oz and the data column x. Not all commands accept both: summary(oz) works but quantile(oz, c(0.7, 0.8, 0.9)) does not work.

11 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Example Sort the ozone measurements in increasing order and determine the sample percentiles each ordered observation corresponds to. Solution: The commands sort(x); 100*(1:length(x) - 0.5)/length(x) return the order statistics and the percentile each order statistic estimates.

12 The Boxplot Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots The five number summary given by the summary command is the basis for the boxplot. A boxplot displays the central 50% of the data with a box: the lower and upper edges are at q 1 and q 3, respectively, a line inside the box represents the median. Extending from each edge of the box are whiskers: The lower (upper) whisker extends from q 1 (q 3 ) until the smallest (largest) observation within 1.5 interquartile ranges from q 1 (q 3 ). Observations further from the box than the whisker ends (i.e., smaller than q IQR or larger than q IQR) are called outliers, and are plotted individually.

13 The R command boxplot Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Example Construct the box plot for the ozone data. Are there any outliers? Solution: The ozone data are already in the object x. Use the command There are two outliers. boxplot(x, col= grey ).

14 Outline Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

15 Pie Charts and Bar Graphs Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Pie charts and bar graphs are used with count data which display the percentage of each category in the sample. For example, counts (or percentages or proportions) of different ethnic or education or income categories, the market share of different car companies, and so on. The pie chart is popular in the mass media and one of the most widely used statistical charts in the business world. It is a circular chart, where the sample is represented by a circle divided into sectors whose sizes represent proportions.

16 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots It has been pointed out that it is difficult to compare different sections of a given pie chart. According to Steven s power law length is a better scale to use than area. The bar graph uses bars of height proportional to the proportion it represents. Remark: When the heights of the bars are arranged in a decreasing order, the bar graph is also called Pareto chart. The Pareto chart is one of the key tools for quality control, where it is often used to represent the most common sources of defects in a manufacturing process, the most frequent reasons for customer complaints, etc.

17 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots R commands for the pie chart and bar graph Example The MarketShareLightVeh data set displays the November 2011 light vehicle market share of car companies. Import the data set into the R data frame lv, and construct a pie chart and bar graph. Solution: The data frame lv has two columns with labels Company and Percent containing the manes of companies and their percent market share, respectively. (You can see that by typing the command lv after importing the data.) The R commands for the pie chart and bar graph are: attach(lv) pie(percent, labels=company, col=rainbow(length(percent))) barplot(percent, names.arg=company, col= rainbow(length(percent)), las=2) detach(lv) The option las=2 in the barplot command is what results in the company names to be written vertically.

18 Histograms Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots In histograms the range of the data is divided into bins, and a box is constructed above each bin. The height of each box is the bin s frequency. Alternatively, the heights can be adjusted so the histogram s area is one. R will automatically choose the number of bins but it also allows user specified intervals. Moreover, R offers the option of constructing a smooth histogram. In stem and leaf plots each observation gets split into its stem, which is the beginning digit(s), and its leaf, which is the first of the remaining digits. They retain more information about the original data but do not offer as much flexibility in selecting the bins.

19 The R data set faithful Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots x = faithful$eruptions # set the eruption duration data in x hist(x) # basic frequency histogram hist(x, freq = FALSE) # histogram area = 1 plot(density(x)) # basic smooth histogram hist(x, freq = F) ; lines(density(x)) # superimposes the two stem(x) # basic stem and leaf plot hist(x, freq = F, col= grey, main= Histogram of Old Faithful eruption durations, xlab= Eruption durations ) ; lines(density(x), col= red )

20 Outline Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

21 Scatterplot with gender identification Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots With the bear measurements data in the data frame br, a basic chest girth and weight scatterplot can be constructed by: attach(br); plot(chest.g, Weight) An enhanced chest girth and weight scatterplot with gender differentiation can be constructed by: plot(chest.g, Weight, pch=21, bg=c( red, green )[unclass(sex)]) legend( x=22, y=400,pch = c(21,21), col = c( red, green ), legend = c( Female, Male ))

22 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots Scatterplot matrix with gender identification For more than two variables, a scatterplot matrix arranges all pairwise scatterplots in a matrix form. With the bear measurements in the data frame br use the command: pairs(br[4:8],pch=21,bg=c( red, green )[unclass(sex)]) # br[4:8] is a data frame consisting of columns 4-8 ( )For a variation, which gives histograms on the diagonal and additional information, use the commands: install.packages( psych ) # installs the package psych library(psych) # it activates the package pairs.panels(br[4:8], pch=21,bg=c( red, green )[unclass(sex)])

23 ( ) 3D Scatterplots Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots With the bear measurements data in the data frame br (use install.packages( scatterplot3d ) if not installed before) use: library(scatterplot3d); scatterplot3d(br[6:8]) # for the basic 3D scatterplot scatterplot3d(br[6:8],angle=35, col.axis= blue, col.grid= lightblue, color= red ) # angle and color controls scatterplot3d(br[6:8], angle=35, col.axis= blue, col.grid= lightblue, color= red, type= h, box=f) # vertical lines, no box scatterplot3d(br[6:8],pch=21,bg=c( red, green )[unclass(br$sex)]) # with gender differentiation detach(br)

24 Output options ( ) Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots The figure can be saved as pdf, or jpg etc. Alternatively: pdf( Desktop/HistOF.pdf ) # saves figure in Desktop/HistOF.pdf hist(x, freq = F, col= grey ); lines(density(x), col= red ) dev.off() # this must be done before opening the pdf file. To save it as a jpg file replace pdf( Desktop/HistOF.pdf ) in the above set of commands by jpeg( Desktop/HistOF.jpg ). To save text output to a txt file, for example the stem and leaf plot, copy and past, or use: sink( Desktop/StemOF.txt ); stem(x); sink(file=null)

25 Randomization, Confounding and Simpson s Paradox Comparative studies aim at discerning and explaining differences between two or more populations. Examples include: The comparison of two methods of cloud seeding for hail and fog suppression at international airports, the comparison of the survival times of a type of root system under different watering regimens, the comparison of the effectiveness of three cleaning products in removing four different types of stains.

26 Randomization, Confounding and Simpson s Paradox Some common terms used in comparative studies are: Experimental units: These are the subjects or objects on which measurements are made. Response variable: The variable being measured. One-factor studies. Factor levels; treatments; populations Multi-factor studies. Factor level combinations; treatments; populations The notions of factor(s), factor levels and factor level combinations are explained in the two examples that follow.

27 Randomization, Confounding and Simpson s Paradox Example To compare the effect of four different watering regimens on the survival times of a type of root system, The roots are the experimental units. The response variable is the survival time. Watering is the factor. The different watering regimens are the factor levels or treatments. Treatments correspond to populations.

28 Randomization, Confounding and Simpson s Paradox Example In the same root survival time as above, it is desired to also study the effect of depth on the survival of the root systems. Two different depths are to be considered. This is now a two-factor study: Factor A is depth with two levels. Factor B is watering with four levels. Treatments, or populations, are the different factor level combinations. There are 2 4 = 8 treatments. As before, the root systems are the experimental units, and the survival time is the response variable.

29 Randomization, Confounding and Simpson s Paradox The following table shows the eight factor level combinations of the above two-factor study: Factor B Factor A Tr 11 Tr 12 Tr 13 Tr 14 2 Tr 21 Tr 22 Tr 23 Tr 24

30 Contrasts Randomization, Confounding and Simpson s Paradox Comparisons of treatments, or populations, typically focus on differences (e.g., of means, or proportions). Such differences are called contrasts. For example, the comparison of two different cloud seeding methods may focus on the simple contrast µ 1 µ 2. In one-factor studies where the factor has more than two levels, a number of different contrasts may be of interest. An example follows. In multi-factor studies interest lies in more specialized contrasts, which are discussed in the section on Factorial Experiments.

31 Randomization, Confounding and Simpson s Paradox Example In a study to compare the mean tread life of four types of high performance tires, possible sets of contrasts of interest are 1 µ 1 µ 2, µ 1 µ 3, µ 1 µ 4 (control vs treatment) 2 µ 1 + µ 2 2 µ 3 + µ 4 2 (brand A vs brand B) 3 µ 1 µ, µ 2 µ, µ 3 µ, µ 4 µ (tire effects)

32 Outline Randomization, Confounding and Simpson s Paradox 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

33 Randomization, Confounding and Simpson s Paradox To avoid comparing apples with oranges, the experimental units for the different treatments must be homogenous. If fabric age affects the effectiveness of cleaning products then, unless the fabrics used in different treatments are age- homogenous, the comparison of treatments will be distorted. To mitigate the distorting effects, or confounding, of other possible factors, called lurking variables, it is recommended that the allocation of units to treatments be randomized.

34 Randomization, Confounding and Simpson s Paradox Randomizing the allocation of fabric pieces to the different treatments (cleaning product and stain) avoids confounding with the factor age of fabric. The distortion caused by lurking variables in the comparison of proportions is called Simpson s Paradox.

35 Randomization, Confounding and Simpson s Paradox Example The success rates of two treatments, Treatments A and B, for kidney stones are: Treatment A Treatment B 78% (273/350) 83% (289/350) The obvious conclusion is that Treatment B is more effective. The lurking variable here is the size of the kidney stone.

36 Randomization, Confounding and Simpson s Paradox Example (Kidney Stone Example Continued) When the size of the treated kidney stone is taken into consideration, the success rates are as follows: Small Large Combined Tr.A 81/87 or /263 or /350 or.78 Tr.B 234/270 or.87 55/80 or /350 or.83 Now we see that Treatment A has higher success rate for both small and large stones.

37 Randomization, Confounding and Simpson s Paradox Example (Batting Averages) The overall batting average of baseball players Derek Jeter and David Justice during the years 1995 and 1996 were and 0.270, respectively. But looking at each year separately we get a different picture: Combined Jeter 12/48 or /582 or /630 or.310 Justice 104/411 or /140 or /551 or.270 Justice had a higher batting average than Jeter in both 1995 and 1996.

38 Factorial Experiments, Main and Interaction Effects Definition A study is called a statistical experiment if the investigator controls the allocation of units to treatments or factor-level combinations, and this allocation is done in a randomized fashion. Otherwise the study is called observational. Causation can only be established via a statistical experiment. Thus, a relation between salary increase and productivity does not imply that salary increases cause increased productivity. Observational studies cannot establish causation, unless there is additional corroborating evidence. Thus, the link between smoking and health has been established through observational studies with the use of additional corroborating evidence.

39 Outline Factorial Experiments, Main and Interaction Effects 1 Basic Statistics and the Boxplot Pie Charts, Bar Graphs, and Histograms Scatterplots, Scatterplot Matrices and 3D Scatterplots 2 Randomization, Confounding and Simpson s Paradox 3 Factorial Experiments, Main and Interaction Effects 4

40 Factorial Experiments, Main and Interaction Effects A statistical experiment involving several factors is called a factorial experiment if all factor-level combinations are considered. Thus, Factor B Factor A Tr 11 Tr 12 Tr 13 Tr 14 2 Tr 21 Tr 22 Tr 23 Tr 24 is a factorial experiment if all 8 treatments are included in the study.

41 Main Effects and Interactions Factorial Experiments, Main and Interaction Effects In factorial experiments it is not enough to consider differences between the levels within each factor separately. Possible synergistic effects are also of interest. Definition If there are synergistic effects among two different factors, i.e., when a change in the level of factor A has different effects on the response depending on the level of factor B, we say that there is interaction between the two factors. The absence of interaction is called additivity.

42 Factorial Experiments, Main and Interaction Effects Example An experiment considers two types of corn, used for bio-fuel, and two types of fertilizer. The following two tables give possible population mean yields for the four combinations of seed type and fertilizer type.

43 Factorial Experiments, Main and Interaction Effects Fertilizer Row Main I II Averages Row Effects Seed A µ 11 = 107 µ 12 = 111 µ 1 = 109 α 1 = 0.25 Seed B µ 21 = 109 µ 22 = 110 µ 2 = α 2 = 0.25 Column Averages µ 1 = 108 µ 2 = µ = Main Column β 1 = 1.25 β 2 = 1.25 Effects Here the factors interact.

44 Factorial Experiments, Main and Interaction Effects Fertilizer Row Main Row I II Averages Effects Seed A µ 11 = 107 µ 12 = 111 µ 1 = 109 α 1 = 1 Seed B µ 21 = 109 µ 22 = 113 µ 2 = 111 α 2 = 1 Column Averages µ 1 = 108 µ 2 = 112 µ = 110 Main Column β 1 = 2 β 2 = 2 Effects Here the factors do not interact.

45 Factorial Experiments, Main and Interaction Effects Under additivity: There is an indisputably best level for each factor, and The best factor level combination is that of the best level of factor A with the best level of factor B. What is the best level of each factor in the above design? Under additivity, the comparison of the levels within each factor are based on the factor s main effects: Under additivity, α i = µ i µ, β j = µ j µ µ ij = µ + α i + β j

46 Factorial Experiments, Main and Interaction Effects When the factors interact, the cell means are not given in terms of the main effects as above. The difference γ ij = µ ij (µ + α i + β j ) quantifies the interaction effect. For example, in the above non-additive design, γ 11 = µ 11 µ α 1 β 1 = = 0.75.

47 Factorial Experiments, Main and Interaction Effects Data Versions of Main Effects and Interactions Data from a two-factor factorial experiment use three subscripts: Factor B Factor A x 11k, x 12k, x 13k, k = 1,..., n 11 k = 1,..., n 12 k = 1,..., n 13 2 x 21k, x 22k, x 23k, k = 1,..., n 21 k = 1,..., n 22 k = 1,..., n 23

48 Factorial Experiments, Main and Interaction Effects Sample versions of main effects and interactions are defined using x ij = 1 n ij n ij x ijk, k=1 instead of µ ij : α i = x i x, βj = x j x Sample Main Row and Column Effects γ ij = x ij (x + α i + β ) j Sample Interaction Effects

49 Factorial Experiments, Main and Interaction Effects Sample versions of main effects and interactions estimate their population counterparts but, in general, they are not equal to them. Thus, even if the data has come from an additive design, the sample interaction effects will not be zero. The interaction plot is a graphical technique that can help assess whether the sample interaction effects are significantly different from zero. For each level of, say, factor B, the interaction plot traces the cell means along the levels of factor A. See CloudSeedInterPlot.pdf for an example. For data coming from additive designs, these traces (or profiles) should be approximately parallel.

50 In this unit we will see the comparative boxplot, the comparative bar graph, and the interaction plot, where The comparative boxplot consists of side-by-side individual boxplots for the data sets from each population. It provides a visual impression of differences in the median and percentiles of the levels in one-factor studies. The comparative bar graph provides visual comparison of the categories proportions for two populations. The interaction plot provides a visual aid for assessing the presence of interactions in two-factor studies. R commands for computing the main effects and interactions for two-factor design will also be given in this unit.

51 The Comparative Boxplot Example Iron concentration measurements from four ore formations are given in the FeData data set. Read this data into the R data frame fe, and construct a comparative boxplot. Solution: With the data set read into the data frame fe, use the commands fe[1:3,] # to see what the data frame looks like boxplot(fe$conc fe$ind, col=rainbow(4))

52 The Notched Boxplot Notched boxplots provide additional information through the notches: If notches do not overlap we may, as an informal test, conclude that the population medians differ. Import the steal strength (SteelStrengthData.txt) and the robot reaction times (RobotReactTime.txt) data in the data frames ss and rt, respectively, use attach(ss); attach(rt), and compare the notched boxplots produced by boxplot(value Sample, col=rainbow(2),notch=t) boxplot(time Robot, col=rainbow(2),notch=t); detach(ss); detach(rt)

53 The Comparative Bar Graph Example The light vehicle market share of car companies for the month of November in 2010 and 2011 is given in the data file MarketShareLightVehComp.txt. Construct a comparative bar graph. Solution: With the data set read into the data frame lv2, use the commands m=rbind(lv2$percent 2010, lv2$percent 2011) barplot(m, names.arg=lv2$company, ylim=c(0,20), col=c( darkblue, red ), legend.text= c( 2010, 2011 ), beside=t,las=2)

54 The Interaction Plot The interaction plot is a useful graphical technique for assessing whether the sample interaction effects are sufficiently different from zero to imply a non-additive design. For each level of one factor, say factor B, the interaction plot traces the cell means along the levels of the other factor. If the design is additive, these traces (also called profiles) should be approximately parallel.

55 Example (Cloud seeding in Tasmania) The could seeding data (CloudSeed2w.txt) was collected to study the effect of the factors seed and season on rainfall (source: Miller, A.J, et al. (1979), Analyzing the results of a cloud-seeding experiment in Tasmania, Communications in Statistics - Theory & Methods, A8(10), ). Construct the interaction plot. Solution: Import the data set into the data frame cs and use attach(cs) and the command interaction.plot(season, seeded, rain, col=c(2,3), lty = 1, xlab= Season, ylab= Cell Means of Rainfall, trace.label= Seeding )

56 Computation of the main and interaction effects of a two-factor design require the use of the tapply command in R, and the commands for computing certain means of a matrix. These commands are presented first.

57 The R function tapply The tapply function is useful when we need to break up a set of numbers into subgroups, which are defined by some classifying factor(s), compute a statistic on each subgroup, and return the results in a convenient form. We will use the tapply function to compute the sample averages in all factor-level combinations in a b designs, i.e., we will break up the set of values of the response variable into the subgroups defined by the factor level combinations, compute the sample mean for each subgroup, and return the results in a matrix form. See the example that follows

58 Example Compute the sample averages for all factor-level combinations of the cloud seeding data set. Solution: With the cloud seeding data set in the data frame cs, use the command mcm=tapply(cs$rain, cs[,c(2,3)],mean) # matrix of cell means mcm # to display the results

59 Means of a Matrix The functions mean, rowmeans, and colmeans, when applies to a matrix, return the mean of all elements of the matrix, the vector of row means and the vector of column means. mean(mcm) # returns the mean of all sample means. rowmeans(mcm) # returns the vector of row means colmeans(mcm) # returns the vector of column means

60 Main Effects and Interactions in R Example Use R to compute the main and interaction effects for the factors seed and season of the cloud seeding data. Solution. Use the following commands: alphas= rowmeans(mcm) - mean(mcm) # main row effects betas = colmeans(mcm) - mean(mcm) # main column effects gammas=t(mcm-mean(mcm)-alphas)-betas # matrix of interaction effects.