MacAnova User's Guide
Contents
1. T T T T T E 30 25 20 4 y 15 D D B D 10 7 D 7 D F c D C c 5H E D c D B E 2 Sa oe a a A A 0 2 3 4 5 6 7 8 9 10 2 62 MacAnova Version 4 04 Cmd gt chplot x run 10 y run 10 3 run 5 ymin 0 title Sample plot by chplot no plotting symbols provided Sample plot by chplot 5 30 z 5 Zor ral 5 207 5 Y 4 LSF 4 5 4 10 4 5 4 5 3 4 5 3 SK 5 4 3 3 4 1 8 3 2 I iL 1 iL 0 1 2 3 4 5 6 7 8 9 10 x 2 15 4 Equally spaced x values The exception to the requirement that x and y have the same number of rows is a convention you can use when you want to plot the rows of y against equally spaced values on the X axis When x is a REAL scalar single number y i is plotted against vector x x 1 x 2 that is they are equally spaced by 1 starting with x When x is a vector of length 2 say vector x0 d y i is plotted against vector x0 x0 d x0 2 d that is equally spaced starting with x0 and incrementing by d For example plot 1 y is equivalent to plot run nrows y y and lineplot vector 1967 1 12 y might be used to plot monthly values for which y 1 was data for January 1967 2 15 5 Graphics keywords See Sec 8 5 for information on the use of keywords xmin ymin xmax ymax xlab ylab xaxis yaxis title file keep and show and on how to add information to previously created grap2. by In addition if the file contains lines all beginning Thus suppose myfile1 dat looks like 2 522 531 602 684 513 603 30 not be read 41 10 0 448 Then vecread myfilel dat stop skip will return exactly the same vector as before If the first line were not skipped the 30 would be read as an item of data Suppose myfile2 dat contains data arranged in columns like the following 600 522 61 10 4 4 533 644 246 595 448 431 Data on 5 variables n 489 481 358 602 892 513 6 640 612 623 684 644 603 You can read it into matrix y as follows Cmd gt y lt matrix vecread myfile2 dat skip quiet F 5 Echoed because of quiet F Data on 5 variables n 6 Cmd gt y 1 1 600 555 2 1 MISSING 644 3 1 522 246 4 1 614 595 5 1 410 448 6 1 MISSING 431 489 481 358 602 892 513 Note keyword phrase quiet F which results in any skipped lines being printed Transposing the output of matrix is necessary because data is read from the file row by row but stored in the computer column by column Thus simply y lt matrix vecread myfile2 dat skip 5 would produce a matrix with 5 rows with each row corresponding to a column of the data in the file 2 32 640 612 623 684 644 603 640 612 623 684 644 603 MacAnova Version 4 04 You can also use vecrea
3. sort x returns a vector or matrix each column of which is the corresponding column of x arranged in increasing or alphabetical order sort x down T orders each column in decreasing or reverse alphabetical order In both cases x is REAL MISSING values are moved to the end rank x computes the ranks of the non missing data in each column of x with the smallest or alphabetically first value assigned rank 1 rank x down T does the same except the largest or alphabetically last value assigned rank 1 For REAL data the rank of a MISSING value is set MISSING By default when there are ties in REAL data the average of the ranks associated with each set of ties is normally computed For CHARACTER data the ranks associated with tied elements are distinct and unpredictable See below for alternative treatment of ties with REAL data grade x computes what might be called inverse ranks of the columns of x When x is a vector after j lt grade x j 1 is the index row number of the smallest or alphabetically first element of x j 2 is the index of the second smallest and so on If length x is n then grade x n is the index of the largest number maximum or last in alphabetical order Thus x grade x is the same as sort x x grade x 1 returns a matrix the same shape as x containing the rows of x rearranged so that the first column is in increasing order This is probably the most
4. 5 6 W7 10 are special and are interpreted as diamond plus sign cross triangle star dot and small cross For example if y is a vector chplot x y 6 is identical to plot x y When symhas a single element as in chplot x y it is used for all points plotted When symis a vector whose length matches the number of columns in y then sym j labels all points associated with column j that is the point y i j is plotted against 2 61 MacAnova Version 4 04 x i with character or number sym j For example if y has 3 columns chplot x y run 3 labels all the points from y 1 y 2 and y 3 with 1 2 and 3 respectively When the symis a vector whose length does not match the number of columns in y then sym j labels row j with the symbols repeating cyclically if length sym lt nrows y Finally when sym is a matrix it must have the same number of columns as y and symbols for plotting the values from row i of y are taken from row i of sym repeating rows of symcyclically if necessary You can omit symaltogether if y has only one column each point is labelled with the row number if y has several columns all the points from a column are labelled with the column number Cmd gt chplot x run 10 y run 10 3 run 5 vector A uy B aa AA ae D ty he y ymin b 0 N title Sample plot by chplot Sample plot by chplot
5. furthest lowest and highest points that are still within the inner fences lower quartile 1 5 IQR and upper quartile 1 5 IQR Points outside the inner fences may be tentatively identified as outliers Points outside the the outer fences lower quartile 3 IQR and upper quartile 3 IQR are plotted as o and and points between the inner and outer fences are plotted as 2 40 MacAnova Version 4 04 Cmd gt boxplot temperaturesSSaturday temperaturesSSunday xX O Roon Bn Sez nogcrad temperaturesSMonday Box Plot I 3 j_ _ 2 n T 50 55 60 65 70 TS 80 85 Values Cmd gt vboxplot temperaturesSSaturday temperaturesSSunday temperaturesSMonday same with vertical orientation Box Plot ii 2 3 Box Number 2 41 MacAnova Version 4 04 The bottom or left box represents the Saturday temperatures the middle box the Sunday temperatures and so on No temperatures were outliers so there are no symbols beyond the whiskers Cmd gt boxplot temperaturesSSaturday temperaturesSSunday temperaturesSMonday vertical T is an alternative way to get vertically oriented boxes If str is a structure whose components are REAL vectors boxplot str draws parallel box plots for each component Thus since temperatures is a structure boxplot tempera
6. important application of grade grade x down T does the same except j 1 is the index of the largest value or last in alphabetical order When x is a matrix grade x processes each column separately If x is REAL the indices of any MISSING values are always put in the last rows of grade x and grade x down T Function rankits computes the normal scores or rankits of the non missing elements in each column and sets the rankit of a MISSING value to MISSING When x is a vector and there are no ties rankits x i is the normal probability point z where p rank x i 375 n 25 When there are ties rankits breaks them arbitrarily Note that the order of the elements of x is retained Function halfnorm is similar to rankits except it computes half normal scores that is when there are no ties halfnorm x i is Zp where pi 5 5 rank abs x i 375 n 25 2 45 MacAnova Version 4 04 Cmd gt x lt matrix vector 2 4 5 8 2 3 4 1 7 8 4 3 6 5 5 X 1 1 2 2 8 2 1 MISSING 3 4 3 1 4 4 3 4 1 5 1 6 5 1 8 7 5 Cmd gt print sort sort x rank rank x rankits rankits x grade grade x WARNING MISSING values in argument to sort WARNING MISSING values in argument to rank WARNING MISSING values in
7. into a separate MacAnova variable You sometimes can use pre defined macro readcols which makes use of vecread to read such file and create REAL vectors one for each column in the file The arguments to readcols are the file name and the unquoted names of the variables into which the columns should be placed Cmd gt readcols myfile2 dat xl x2 x3 x4 x5 skip Data on 5 variables n 6 2 33 MacAnova Version 4 04 This creates REAL vectors x1 x2 x3 x4 and y containing the data from columns 1 through 5 Cmd gt list x1 x2 x3 x4 x5 x1 REAL 6 x2 REAL 6 x3 REAL 6 x4 REAL 6 x5 REAL 6 Cmd gt xl column variable 1 1 600 MISSING 522 614 410 6 MISSING If the file has a different number of columns from what you expect readcols usually prints an error message Cmd gt readcols myfile2 dat x1 x2 x3 x4 skip try 4 cols Data on 5 variables n 6 ERROR number of rows must divide length of data However if the number of data elements in the file is actually divisible by the number of variable names you provided there will be no error message but the variables set will be of the wrong length and will not correspond to columns in the file Cmd gt readcols myfile2 dat xl x2 x3 x4 x5 x6 skip Data on 5 variables n 6 Cmd gt list x1 x1 REAL 5 Output from list Cmd gt xl x1 has length 5 not 6 1 600 644 504 602 644 Macro
8. 11 Leaf digit unit 1 Keyword phrase depth F suppresses printing the depth column and keyword phrase stat T prints the extremes and the quartiles Cmd gt stemleaf temperaturesSSaturday 5 depth F stat T n 10 Min 59 Q1 69 M 76 5 Q3 89 Max 93 9 59 158 69 13 1 1 represents 11 Leaf digit unit 1 WO OnwAID UI A histogram is more conventional way to display the distribution of data points You can draw histograms using pre defined macro hist If you are satisfied to let 2 43 MacAnova Version 4 04 MacAnova pick the number of bars in the histogram you can simply type hist x where x is a REAL vector Like stemleaf an optional second argument specifies the number of bars The heights of the bars is in the so called density scale that is the product of the width of a bar by its height is the proportional of values falling in the bar Cmd gt hist temperaturesSMonday 6 draw histogram with 6 bars Histogram of temperatures Monday with total area 1 ae a e O E e 50 55 60 65 70 75 80 85 temperatures Monday You can in fact completely specify the class limits For example Cmd gt hist temperaturesSMonday vector 40 50 60 70 80 85 90 draws a histogram with unequally spaced class limits 40 50 60 70 80 85 and 90 You can specify a title and X axis labels for a histogram or for a box plot using key
9. 2 14 More on rep and run Functions run and rep are very useful for setting up factor variables in ANOVA models described in Chapter 3 As described in Sec 2 8 12 run generates equally spaced sequences and rep replicates its first argument We saw there that run n returned vector 1 2 n and rep v n returned vector v v v Where there are n repetitions and where v is a scalar or vector Cmd gt rep run 4 3 3 replicates of 1 2 3 4 1 1 2 3 4 6 2 3 4 11 3 4 There is a more general usage of rep In rep v counts where counts is a vector of non negative integers of the same length as v the corresponding v i is repeated counts i times In particular rep v rep m n where n is the length of v produces m copies of each element of v Cmd gt rep run 3 rep 4 3 4 copies of each element of run 3 1 1 1 1 1 6 2 2 2 3 11 3 3 Thus you might use rep run 4 3 and rep run 3 rep 4 3 to generate treatment number and block numbers associated with a randomized block design with 4 treatments and 3 replicates Cmd gt rep run 4 vector 2 1 0 6 2 1 s 1 2 no 3 s and 6 4 s 1 1 1 2 4 4 6 4 4 4 4 2 58 MacAnova Version 4 04 2 15 Making graphs MacAnova has several functions and macros for making graphs of data The basic functions are as follows Function or Macro plot x y Description Scatter plot or impulse plot of y versus x point
10. 2 degrees of freedom When the variances cannot be assumed equal you 2 2 so sS can use the unpooled standard error s _ Unfortunately Student s t n M distribution is no longer directly applicable However a good approximation is to use v v Student s t with estimated degrees of freedom f where v s n i 1 Vi V2 Rol neal 2 Snedecor and Cochran 1980 Sec 6 11 Including pooled F as an extra argument to t2val and t2int directs MacAnova to use this approximation For t2val the result returned is a structure with components t and df Cmd gt t2int x1 y 95 pooled F slightly different from before 1 4 1205 0 85676 Cmd gt t2val x1 y pooled F not assuming same variances component 1 3 2186 Value of t statistic component df 1 16 926 Estimated degrees of freedom 2 12 7 P values and cumulative distribution functions MacAnova has several functions for computing the cumulative distribution functions CDF of standard probability distributions You can use them to compute P values associated with many standard test statistics The arguments to these functions are always a value x of the random variable of interest followed by the value of parameters as indicated in the following table 2 51 MacAnova Version 4 04 Distribution Usage Standard normal cumnor x Student s t on df deg
11. 60 70 or run 30 70 10 class limits Cmd gt bin x b group x using boundaries b component boundaries 1 30 40 50 60 70 component counts 1 2 10 10 3 The first argument x contains the data to be grouped the second b is a vector of class limits Here b defines the boundaries of 4 length b 1 classes or groups Group 1 consists of all values between b 1 and b 2 group 2 consists of all values between b 2 and b 3 and soon The data are entered in sorted order only to make it easier to check that bin is doing what it should The result is a structure with components boundaries and counts Component boundaries is identical with b while counts j is the number of input values in 2 55 MacAnova Version 4 04 class j that is the frequency of the class Here there are two values 33 and 35 in class 1 ten values 41 41 42 44 47 49 49 50 50 50 in class 2 and so on By default any value exactly on a class limit at the right or upper end of a class is counted in that class so for example the three 50 s are considered to be in class 2 together with 41 through 49 and 60 goes in class 3 Any value not in any class here lt 30 or gt 70 is not counted but a warning message is printed You can suppress the warning message by including silent T as an argument Cmd gt bin vector x 25 75 b data vector enlarged by 2 values WARNING 1 low and 1 high values not counted by bin compone
12. 8 67 7 28 7 35 9 32 1 28 5 28 1 5 Cmd gt w 3 columns 1 1 45 5 58 4 28 7 2 1 42 1 74 1 3549 3 1 53 28 72 82 5 1 4 1 48 5 63 8 2845 5 1 44 5 67 7 28 1 Cmd gt cor w 3 by 3 matrix 1 1 1 0 049932 0 16287 2 1 0 049932 1 0 78611 3 1 0 16287 0 78611 1 If x1 x2 are vectors or matrices all with the same number of rows then cor x1 x2 x3 is equivalent to cor hconcat x1 x2 x3 see Sec 2 10 6 Cmd gt makecols w wl w2 w3 See Sec 2 11 3 Cmd gt cor wl w2 w3 same as cor w 1 1 1 0 049932 0 16287 2 1 0 049932 1 0 78611 3 1 0 16287 0 78611 1 You can use subscripts Sec 2 8 14 to get the correlation between two variables as a single number Cmd gt cor wl w2 1 2 compute a single correlation coef 1 1 0 049932 If there are any MISSING values in the arguments correlations are computed using only complete cases effectively deleting any row with any MISSING values Cmd gt ww lt w ww 2 3 lt put a MISSING value in row 2 2 49 MacAnova Version 4 04 Cmd gt cor ww same as cor w 2 WARNING 1 cases with missing values deleted in cor 1 1 1 0 64695 0 92817 2 1 0 64695 1 0 65979 3 1 0 92817 0 65979 1 2 12 6 Student s t related functions There are four functions for statistical analyses based on Student s t Functions tval and t2val calculate one and two sample t statistic
13. All ranks r r 1 r k 1 k r k 1 2 minimum All ranks r ignore r r 1 r 2 r k 1 in an unpredictable order rankits x ties method and halfnorm x ties method are equivalent to finding the normal probability points corresponding to p rank x ties method i 375 n 25 and p 5 5 rank abs x ties method i 375 n 25 respectively It is hard to imagine a situation in which ties minimum would be useful with rankits and halfnorm Cmd gt rank vector 3 2 2 1 3 ties average default 1 Ais 2d 2 5 1 4 5 Cmd gt rank vector 3 2 2 1 3 ties minimum 1 4 2 2 1 4 Cmd gt rank vector 3 2 2 1 3 ties ignore 1 4 2 3 1 5 Cmd gt rankits vector 3 2 2 1 3 ties average 1 0 79164 0 24104 0 24104 1 1798 0 79164 Cmd gt rankits vector 3 2 2 1 3 ties ignore default 1 0 4972 0 4972 0 1 1798 1 1798 2 12 4 sum prod max min These functions compute summary values for a REAL or LOGICAL vector or for each column of a REALor LOGICAL matrix For LOGICAL data True is interpreted as 1 and False as 0 Cmd gt print sum sum x prod prod x min min x max max xX WARNING MISSING values found by sum WARNING MISSING values found by prod WARNING MISSING values found by min WARNING MISSING values found by max sum 1 1 19 17 26 Su
14. DATAPATHS Cmd gt adddatapath timeser DATAPATHS add timeser at start 1 timeser 2 Macintosh HD MacAnova Folder Data 3 Macintosh HD MacAnova Folder Cmd gt adddatapath mvdata T DATAPATHS add mvdata at end 1 timeser 2 Macintosh HD MacAnova Folder Data 3 Macintosh HD MacAnova Folder 4 mvdata The above uses the Macintosh way of separating names in a path using On other computes you would use names like D MACANOVA DATA DOS Windows or users kb macanova data Unix Variable HOME comes into play when you use a file name of the form name on a Macintosh or name on other computers MacAnova looks for the file in the directory whose name is in HOME Cmd gt HOME 1 Macintosh HD MacAnova Folder Cmd gt y lt vecread Data Hald found You can change HOME to be the name of any folder or directory Cmd gt HOME lt Macintosh HD MacAnova Folder Data Cmd gt y lt vecread Hald found 2 12 Simple statistics While MacAnova is oriented towards the analysis of variance multivariate analysis and time series analysis it provides many simpler statistical functions as well 2 12 1 describe When you want simple descriptive statistics of data in variable x use describe x This computes for each column of x the number of non missing values the mean variance median maximum minimum and upper and
15. Studentized range statistic based on 5 samples of size 20 You can actually compute a page of an F table with a single command The following computes the upper 5 point of F for 1 through 8 numerator degrees of freedom and 1 through 5 denominator degrees of freedom Cmd gt invF 1 05 run 8 rep 1 5 rep 1 8 run 5 8 1 1 161 45 8 513 10 12 7 7086 6 6079 2 1 TISS 19 9 S52 6 9443 5 7861 3 1 215 71 19 164 9 2766 6 5914 5 4095 4 1 224 58 19 247 9 1172 6 3882 5 1922 5 1 230 16 19 296 9 0135 6 2561 5 0503 6 1 233 99 1 9 33 8 9406 6 1631 4 9503 7 1 236 77 19 353 8 8867 6 0942 4 8759 8 1 238 88 19 371 8 8452 6 041 4 8183 2 12 9 Grouping data in class intervals bin An important method of summarizing a data set is to compute a grouped frequency distribution by counting the number of values that lie in a set of class intervals defined by class boundaries b4 lt b lt b3 lt by lt by 1 Class interval j consists of all values between b and b Depending on who is making the definition either the left end b or the right end b of the interval is considered in the interval The counts in each interval are the frequencies Function bin allows you to make this summary It is probably best explained by example Cmd gt x lt vector 33 35 41 41 42 44 47 49 49 50 50 50 51 53 53 54 54 54 55 57 58 60 61 62 67 enter sorted data Cmd gt b lt vector 30 40 50
16. Xmin S the range from a normal sample of size k with standard deviation o and s is an independent estimate of o with df degrees of freedom Itis often applied with data from k independent samples each of size n with R s max min Spoolea Vn where Said is the error mean square from an analysis of variance 2 53 MacAnova Version 4 04 Ztd Non central t is the ratio where z is standard normal independent of X in F Xf 2 the denominator Non central XA with non centrality is Ge 6 where z4 i 1 f Zo Zf are independent standard normal and yi 8 Non central i 1 2 x E ne where the numerator and denominator are independent Non X 8 x A central B A f 20 g 2y where the two x s are independent i LA t X 2 12 8 Probability points and inverse cumulatives MacAnova can compute probability points or critical values inverse cumulatives for many of the distributions in Sec 2 12 7 Most require you to specify degrees of freedom or shape parameters The arguments to these functions are always the probability value of interest followed by distribution parameters as indicated in the following table Distribution Usage Standard normal invnor p Student s t on df degrees of freedom invstu p df x2 on df degrees of freedom invchi p df Non central y2 on df degrees of freedom and invchi p df lambda non cen
17. argument to rankits WARNING MISSING values in argument to grade sort 1 1 2 1 3 2 1 4 2 4 3 1 5 3 5 4 1 8 4 6 5 1 MISSING 7 8 MISSING at end rank 1 1 1 2 5 2 1 MISSING 3 2 3 1 2 4 1 4 1 3 1 4 5 1 4 5 3 rankits 1 1 1 0491 0 4972 1 1798 2 1 MISSING 0 0 4972 3 1 0 29931 0 4972 1 1798 4 1 0 29931 Sted 98 0 4972 5 1 1 049 1 1798 0 a Ciel 1 4 3 2 1 3 1 2 3 1 4 2 5 4 1 5 3 4 5 1 2 5 1 Note the use of keywords sort rank rankits and grade in print to provide labels for the output See Sec 7 4 1 Without them all four matrices would be labeled MATRIX Cmd gt print downsort sort x down T downrank rank x down T downgrade grade x down T same with down T WARNING MISSING values in argument to sort WARNING MISSING values in argument to rank WARNING MISSING values in argument to grade downsort 1 1 8 7 8 2 1 5 4 6 3 1 4 3 5 4 1 2 2 4 5 1 MISSING 1 3 2 46 MacAnova Version 4 04 downrank 1 1 4 4 il 2 1 MISSING 3 4 3 1 3 2 5 4 1 2 5 2 5 1 1 a 3 downgrade 1 1 5 5 1 2 1 4 3 4 3 1 3 2 5 4 1 al HE 2 5 1 2 4 3 With any of these functions if x is an array the result is an array of the same size and shape with the function being applied to every vector specified by fixed values of subscripts 2 3 Cmd gt ary lt array vect
18. display current seeds see Sec 2 13 Seeds are 1657851609 and 960912167 1 1 6579e 09 9 6091e 08 Cmd gt x lt runi 5 x this will be transcribed on spool txt 1 0 67374 0 13423 0 82378 0 89615 0 66544 Cmd gt spool suspend spooling Spooling on spool txt suspended Cmd gt x 1 this line and following output will be transcribed 1 1 6737 1 1342 1 8238 1 8962 1 6654 In the windowed versions of MacAnova it is also possible directly to save the contents of the command window to a file using Save Window or Save Window As on the File menu See Appendices B D and F 2 17 Using save and restore to preserve work between sessions All the variables and macros you are currently using are referred to collectively as the workspace During a MacAnova run the workspace is in computer memory RAM and not in a file on disk When you quit MacAnova unless you take precautions your workspace and often the result of a lot of work is normally lost You can save everything in your workspace on 2 65 MacAnova Version 4 04 disk by save fileName where fileName is a quoted string or CHARACTER variable specifying the name of the file On a later MacAnova session restore fileName restores the workspace to the way it was when saved As usual in windowed versions the empty file name lets you select a file Cmd gt x show that vector x is defined 1 0 67374 0 13423 0 82378 0 89615 0 66544 Cmd gt save sessionl
19. easier is simply bin x for which bin selects both the number of classes and the width 2 56 MacAnova Version 4 04 Cmd gt bin x component boundaries 1 32415 38 383 44 617 50 85 57 083 6 63 317 69 55 component counts 1 2 4 6 8 4 6 1 You can use keyword phrase leftendin T with all these usages If x is a REAL matrix rather than a vector each column is grouped all with the same class limits Component boundaries is as before but now component counts is a matrix with each column containing the frequencies for the data in the corresponding column of x 2 13 Random numbers runi and rnorm MacAnova has functions for generating standard normal and uniformly distributed pseudo random numbers runi n returns a vector of n random variables that are uniformly distributed between O and 1 n must be a positive integer rnorm n returns a vector of n random variables which have standard normal u 0 o 1 distribution 2 13 1 Seeding runi and rnorm setseeds and getseeds The values computed by runi and rnorm depend on two internal positive integer seeds which are updated every time a number is computed If the seeds are the same on two different occasions you will get exactly the same sequence of random numbers You can set the seeds by set seeds vector seed1 seed2 or setseeds seed1 seed2 where seed1 and seed2 are positive integers between 1 and 2147483399
20. freedom need not be integers For cumbin n must be a positive integer and p must be between 0 and 1 For all functions when x is a vector or matrix the probability is computed for each element of x producing a vector or matrix Similarly the parameter arguments may be vectors or matrices The sizes and shapes of the arguments must match except that x or any of the parameters may always be a scalar These functions can be used to compute P values based on the observed value x of a test statistic X The lower tail probability P X lt x that each function computes can be directly interpreted as a P value for a one tail test that rejects for values in the left tail of the test statistic Because many standard tests such as x and F are upper tail tests that reject the null hypothesis only for large positive values you often need an upper tail P value P X 2 x 1 P X lt x4 You get this by subtracting the result 2 52 MacAnova Version 4 04 from 1 For example 1 cumchi 15 3 10 computes the P value for a x statistic on 10 degrees of freedom with observed value 15 3 Of course for the binomial and Poisson distributions which are discrete when xp is an integer P X 2 xops 1 P X lt X obs 7 1 To compute a two tail P value for a z or t test use 2 1 cumnor abs xobs or 2 1 cumstu abs xobs df You can compute a two tail P value for an F test of equal variance based on estimated variances var1 and
21. if desired once MacAnova has started up perhaps in a start up file See Sec 7 8 In particular you can add other directory or folder names to DATAPATHS making it a CHARACTER vector HOME and DATAPATHS make it easier for MacAnova to find files If a file name is specified as a simple file name such as halddata with no special path characters such as or Macanova first attempts to read it in whatever the current default directory or folder is If it is not successful then it makes an attempt in directory DATAPATHS 1 if not successful there it looks in DATAPATHS 2 and so on giving up only if it is not found in any directory or folder in DATAPATHS Here is an example of its use File Haldis not in the default directory folder or in DATAPATHS 1 2 37 MacAnova Version 4 04 but is in a sub directory or folder Cmd gt DATAPATHS one folder name in DATAPATHS 1 Macintosh HD MacAnova Folder Cmd gt y lt vecread Hald can t find file Hald there ERROR vecread cannot open file Hald Cmd gt DATAPATHS lt vector Macintosh HD MacAnova Folder Data DATAPATHS add new folder at start of DATAPATHS Cmd gt y lt vecread Hald y run 5 Found 1 7 26 6 60 78 5 Vector DATAPATHS can be of any length providing lots of places to look for a file Pre defined macro adddatapath makes it easier to add a folder or directory name to either the beginning or end of
22. impulse T make impulse plot a 4 qt 4 1 2 3 4 5 7 xstuff 2 60 MacAnova Version 4 04 2 15 1 lineplot Lineplot x y draws lines between the successive points plotted but does not draw any symbol at the points Normally the elements of x should be in monotonic increasing or decreasing order lineplot uses different line types solid dashed dotted for successive columns of y You can use keyword linetype to change the default line types See Sec 8 5 1 Cmd gt lineplot x run 10 y run 10 3 run 5 title Sample line plot Note keywords x and y to label axes Sample line plot 30F g 25 2 0 a i5 4 2 15 1 chplot chplot x y sym where symis a REAL or CHARACTER vector or matrix specifying symbols or characters to be plotted is similar to plot x y When symisa matrix it must have the same number of columns as y When symis REAL its elements must be integers between 1 and 999 and the numbers themselves are used as the plotting symbols When symis a CHARACTER vector or matrix its elements are used as plotting symbols actually only the first three characters are used if there are more For example chplot run 5 run 5 2 vector Cat Dog Wolf Sheep Bird labels the successive points with Cat Dog Wol She and Bir Character symbols 1 2 3 4
23. setseeds 0 0 initializes the seeds with values based on the date and time providing a more or less random starting point for random numbers If you don t explicitly set the seeds the first time random numbers are computed the seeds are initialized as if you had typed set seeds 0 0 You can retrieve current values of the seeds in a vector of length 2 by function getseeds If getseeds is vector 0 0 the seeds have not been set Resetting the seeds to values previously retrieved makes it possible to generate the same sequence of random numbers more than once Here are some simple examples Cmd gt runi 5 5 uniforms without initializing WARNING starting random number seeds are 1979189978 and 1730035780 1 0 2765 0 046009 0 27089 0 21016 0 16606 Cmd gt rnorm 5 5 normals 1 1 9504 0 23464 0 58285 0 43007 0 09322 Cmd gt setseeds 1979189978 1730035780 reset seeds Cmd gt runi 5 same vals 1 0 2765 0 046009 0 27089 0 21016 0 16606 2 57 MacAnova Version 4 04 Cmd gt getseeds value is invisible output is side effect Seeds are 1849669167 and 1493059652 By using runi n as argument p in the inverse cumulative distributions functions described in Sec 2 12 8 you can easily generate pseudo random samples of Student s t F 2 gamma and beta distributed random variables Cmd gt f lt invF runi 5 10 30 f small random sample from F 10 30 1 0 72673 1 6098 2 4452 0 46853 0 76309
24. 9 15 52 74 3 Obs 3 56 8 20 04 3 Obs 4 31 8 47 87 6 Obs 5 7 52 6 33 95 39 Obs 6 55 9 22 09 2 Obs 7 3 71 17 6 02 7 Obs 8 oul 22 44 T25 Obs 9 2 54 18 22 93 1 Obs 10 2 47 4 26 5 9 Obs 11 40 23 34 83 8 Obs 12 66 9 12 ie De ns Obs 13 10 68 8 12 109 4 2 11 4 getdata Macro get data is designed make it easier to work with libraries of data sets in the form readable by matread It has a single unquoted argument that is interpreted as the name of a data set on the file whose name is the value of CHARACTER variable DATAFILE When MacAnova is started up DATAFILE is pre defined to be macanova dat but you may assign a new value at any time getdata uses matread to read the file For example using the default value of DATAFILE Cmd gt hald lt getdata halddata would be equivalent to hald lt matread macanova dat halddata To use getdata to retrieve treedata from file data txt you would need the following Cmd gt DATAFILE lt data txt x lt getdata treedata Subsequent use of getdata would continue to retrieve data from data txt If the file named in DATAFILE is not in the default directory or folder or in one of the directories or folders specified by CHARACTER vector DATAPATHS see Sec 2 11 6 it should be a complete path name such as C MACANOVA MACANOVA DAT or Macintosh Hard Disk MacAnova MacAnova dat See Appendices B through F Note that on DOS Wind
25. A A OFER nt sce Baier Beeb acess Patented Poi ee ees PANES ass Gah seis EE eligi tae stare sweeter josey alge Coe 2 3 4 5 6 7 8 9 10 x In the Macintosh and Windows versions of MacAnova you can copy graphs to the Clipboard and then paste them into word processor or graphic editing documents and you can print them using Print Graph on the File menu see Appendices B and D 2 16 Using spool to save output You often want to keep track of exactly what you have done and what were the results Command spool allows you to spool your session that is save in a file a transcript of what you typed and what MacAnova printed Spooling is a little bit like using a tape recorder everything wanted or unwanted is recorded After leaving MacAnova you can edit the file into a report using a word processor or simply print it as is spool fileName where fileName is a quoted string or CHARACTER variable containing a legal file name starts spooling on the named file From that point on until you stop or suspend spooling everything you type and everything MacAnova responds except high resolution graphs is written to the file in plain text ASCII form In a windowed version Macintosh Windows or Motif if fileName is the empty name you will be able to specify the file using a dialog box On a Macintosh selecting Spool Output To File on the File menu is another way to start spooling 2 64 MacAnova Version 4 04 If the spool f
26. This file consists of the second part of Chapter 2 of MacAnova User s Guide by Gary W Oehlert and Christopher Bingham issued as Technical Report Number 617 School of Statistics University of Minnesota March 1997 describing Version 4 04 of MacAnova This manual is Copyright 1997 Gary W Oehlert and Christopher Bingham all rights reserved Fonts used in this chapter are Palatino Courier and Symbol For information concerning MacAnova write University of Minnesota Department of Applied Statistics 352 Classroom Office Building 1994 Buford Avenue St Paul MN 55108 6042 MacAnova Version 4 04 2 11 Reading data from a file Although you can type small data sets directly into MacAnova using vector and matrix itis often more convenient to read data from a file on your hard or floppy disk The file might have been created in a word processor such as Microsoft Word a text editor such as Edit in DOS or by a spread sheet or data base program MacAnova can read data only from plain text files type TEXT on the Macintosh A plain text file sometimes called an ASCII file is one with no additional information such as font or point size Hence if you use a word processor to create or edit data files to be read by MacAnova it is essential that they be saved as Text or ASCII files In some programs such as Microsoft Word you may have to click on a File Format button to display choices in others the options may be displayed in the
27. d One of the data sets is halddata containing data that has often been used as an example when demonstrating regression techniques Cmd gt hald lt matread macanova dat halddata halddata 13 5 format Hald data from A Hald Statistical Theory with Engineering Applications Wiley New York 1952 p 647 Col 1 X1 percent tricalcium aluminate Col 2 X2 percent tricalcium silicate Col 3 X3 percent tetracalcium alumino ferrite Col 4 X4 percent dicalcium silicate Col 5 Y cumulative heat evolved from cement hardening after 180 days calories gm The lines printed by mat read are the name line and several additional header lines starting with comment lines which come before the data You can suppress the printing of these lines by including quiet T as an additional argument to mat read Since some commands work only with data vectors rather than matrices once you have read in a matrix using matread you may want to split it up into separate vectors each containing a column of the matrix Pre defined macro makecols does exactly this For example you can create vectors from data by Cmd gt makecols hald x1 x2 x3 xX4 y will create vectors x1 x2 x3 x4 and y from the 5 columns of hald You can even combine mat read and makecols ina single expression to create variables Cmd gt makecols matread macanova dat
28. d to read CHARACTER data see also Sec 7 2 Suppose file labels txt contains Age Length Height Width Strength labels for data Then vecread reads a vector of length 6 Cmd gt labels lt vecread labels txt char T labels 1 Age 2 Length 3 Height 4 Width 5 we 6 Strength Note that two successive commas are taken to delimit an empty string You can use keyword phrase silent T to suppress any warning messages Thus if the file starship txt looks like Troy 342 67 Tasha 546 53 Beverly 331 49 Cmd gt y lt matrix vecread starship txt silent T 2 y 1 1 342 67 2 1 546 53 3 1 331 49 returns the 3 by 2 data matrix without commenting on the row labels An alternative method of coping with unreadable items is to use keyword phrase badvalue val where val is a REAL scalar or Any item that does not appear to be a number is read as if it had this value Cmd gt y lt matrix vecread starship txt badvalue 99 3 y 1 1 99 342 67 2 1 99 546 53 3 1 Ho 3 3 1 49 The three names could not be read and were replaced by 99 This has the advantage that you can test to see which items were not readable See Sec 7 3 for information on using vecread to read CHARACTER data 2 11 2 readcols When a data file such as myfile2 dat contains data on several variables the data for each case on a separate line you sometimes want to read each variable column in the file
29. dialog box brought up by Save or Save As If you do not choose a Text format MacAnova will probably not be able to read the file The MacAnova functions for reading data from a plain text file are vecread and matread In addition there are two pre defined macros readcols and getdata All commands which read from a file must have the file name as their first argument This must be a quoted string or CHARACTER variable giving the name of the data file In the windowed versions Macintosh Windows and Motif if the file name is that is it consists of two adjacent quotation marks you are presented with a dialog box with a scrolling list of files from which you can select the file All the commands which read from a text file including vecread matread and macroread can also read from MacAnova CHARACTER variables using keyword phrase string CharVar as first argument instead of the file name See Sec 8 3 2 for details 2 11 1 vecread Function vecread creates a REAL vector from unstructured numerical data in a file The numbers should be separated by blanks or tab characters or may be on separate lines One or more successive question marks in the file is read as MISSING as are isolated periods and asterisks A exclamation point is a stopping character and terminates the read Any other non numeric characters such as letters commas or slashes will be ignored except possibly for printing a
30. eaves and represent the next digit of each number after the stem Because the values of Monday temperatures range only from 49 to 85 here MacAnova uses half digit stems whose leaves are either 0 1 2 3 or 4 for stems labelled 5 6 or 5 6 7 8 or 9 for stems labelled 4 Shean You can control the number of stems stemleaf will use by an optional second argument which allows you to set the maximum number of stems Cmd gt stemleaf temperaturesSMonday 5 use at most 5 stems 9 19 459 158 13445 O1W WWE e OHA UsA represents 11 Leaf digit unit 1 The first column printed contains the depth cumulative counts from each tail of the distribution plus in parentheses the number of leaves on the stem which includes the median Thus there are 5 cases with values 80 and above 6 with values less than 70 and 3 values in the 70 s The leaves are always computed by rounding toward 0 so that a temperature of 79 9 would show up as a leaf of 9 on stem 7 and would not be rounded up to 80 leaf of 0 on stem of 8 When there are outliers defined as values beyond a box plot s inner fences stemleaf normally lists them separately unless outliers F is an argument Let s make the first Monday temperature an outlier Cmd gt monday lt temperaturesSMonday Cmd gt monday 1 lt 120 stemleaf monday 5 9 al 4 2 59 5 6 459 3 7 158 6 8 13445 High 120 1 1 represents
31. halddata quiet T X1 X2 X3 X4 y Ordinarily it is considered an error when mat read cannot find a data set this is considered to be an error If you include keyword phrase not foundok T as an argument no error will be reported and the value NULLis returned This can be useful in writing macros see Sec 9 3 since it allows you to test whether a data set is available on a file Cmd gt x lt matread macanova dat appledata no appledata in file ERROR dataset or macro appledata not found on file macanova dat Cmd gt print x ERROR argument 1 x to print is undefined Cmd gt x lt matread macanova dat appledata notfoundok T Cmd gt print x X NULL Coordinate labels see Sec 8 4 are not part of the file format of data sets readable by matread See Sec 7 1 and thus vectors matrices and arrays read by matread do not normally have such labels You can however add labels using mat read 2 35 MacAnova Version 4 04 keyword labels Here is an example reading halddata echoing of the header lines is suppressed by quiet T Cmd gt data lt matread macanova dat halddata quiet T labels structure Obs vector TricalcAlum TricalcSi TetrcalAlFe DicalcSi CumulHeat Cmd gt print data nsig 4 nsig 4 means print 4 signif digits data TricalcAlum TricalcSi TetrcalAlFe DicalcSi CumulHeat Obs 1 7 26 6 60 1835 Obs 2 2
32. hs You can draw lines between points plotted by plot or chplot by using keyword phrase lines T On all three commands you can use impulses T add impulses to the plot On plot but not on chplot this suppresses the plotting symbols 2 15 6 colplot and rowplot These are pre defined macros that use chplot to draw interaction plots of data in a matrix x colplot x draws line plots of each column of x versus the row numbers run nrows x Data points are labelled with the column number Similarly rowplot x draws line plots of each row of x versus the column numbers run ncols x See Sec 10 12 for an example of the use of colplot 2 63 MacAnova Version 4 04 You can use keyword phrase dumb T as an argument with any plotting command as in plot x y dumb T This results in a crude plot made up of printable characters being printed For many purposes this is sufficient to get a sufficiently clear picture of the relationships being plotted and has the advantage that it is printable on any printer and is written along with other output to a spool file see Sec 2 16 Cmd gt chplot x run 10 y run 10 3 run 5 vector A Me B ee NC Me D Mig he F ymin 0 title Sample dumb plot by chplot dumb T Sample dumb plot by chplot E 30 4 E 25 E 20 E y D 5 E D E D 10 D q E D Cc Cc D D Cc Cc 5 E Cc Cc B B4 E C C B B B B
33. ile already exists spoo1 normally starts writing at its end This allows you to accumulate output from several runs in a single file If this is not what you want use spool fileName new T which starts transcription at the start of the file destroying any information already there To suspend spooling simply type spool with no argument On a Macintosh you can select Stop Spooling on the File menu If spooling has been suspended spool with no argument restarts it On a Macintosh you can select Resume Spooling on the File menu For example you might suspend spooling while you experiment with different analyses of data and then resume it once you have a better idea of what you plan to do Cmd gt spool spool txt start spooling on file spool txt Cmd gt getseeds display current seeds see Sec 2 13 Seeds are 1657851609 and 960912167 1 1 6579e 09 9 6091e 08 Cmd gt x lt runi 5 x this will be transcribed on spool txt 1 0 67374 0 13423 0 82378 0 89615 0 66544 Cmd gt spool suspend spooling Spooling on spool txt suspended Cmd gt x 3 this will not be transcribed on spool txt 1 2 3263 2 8658 2 s L762 2 1038 2 3346 Cmd gt spool resume spooling This line will not be transcribed Resume spooling on spool txt Cmd gt x 1 this line and following output will be transcribed 1 1 6737 1 1342 1 8238 1 8962 1 6654 This is what ends up in file spool txt Cmd gt getseeds
34. lower quartiles The output of describe is a structure with components n min q1 median q3 max 2 38 mean and var MacAnova Version 4 04 Cmd gt x lt matrix vector 2 4 5 8 2 3 4 1 7 8 4 3 6 5 5 xX 1 1 2 2 8 2 1 MISSING 3 4 3 1 4 4 3 4 1 5 1 6 5 1 8 T 5 Cmd gt describe x WARNING missing values in input to describe WARNING output reflects nonmissing data only component n Number of non missing values 1 4 5 5 component min Minimum value 1 2 1 3 component ql Lower quartile 1 3 2 4 component median 2nd quartile 1 4 5 3 5 component q3 Upper quartile 1 6 55 4 6 component max Maximum value 1 8 7 8 component mean Average 1 4 75 3 4 Disk component var Variance 1 6 25 5 3 Bef If you don t want all the statistics you can specify the ones you want using keywords matching the component names Cmd gt describe x mean T var T WARNING missing values in input to describe WARNING output reflects non missing data only component mean 1 4 75 3 4 5 2 component var 1 6 25 5 3 337 If you specify only one statistic describe x mean T you get a scalar or a vector not a structure You can also use keyword phrase a11 T as in describe x all T mean F var F which computes all the statistics except the mean and variance If x is a structure describe x computes the statistics for each component In tha
35. ms down columns prod 1 1 320 168 2880 Product down cols min 1 1 2 HE 3 Minimum value in col max 1 1 8 7 8 Maximum value in col Cmd gt sum x lt 4 count the numbers lt 4 in each column WARNING comparison with missing value s near sum x lt 4 WARNING MISSING values found by sum 1 1 2 4 2 When there are no missing values sum x nrows x computes the column means of a REAL matrix x producing a row vector With any of these functions if the argument is an array the result is an array with the 2 48 MacAnova Version 4 04 same number of dimensions but with the first dimension having length 1 The function is applied to every vector specified by fixed values of subscripts 2 3 We illustrate this use with the array ary created in Sec 2 12 3 Cmd gt sum ary sum over first dimension 1 1 1 80 1 87 4 1 2 1 75 8 91 8 Cmd gt max ary maximimum over first dimensions 1 1 1 32 32 4 1 2 1 28 8 31 4 These functions also accept a list of REAL or LOGICAL scalars or vectors as arguments For example sum 1 3 5 is equivalent to sum vector 1 3 5 2 12 5 Computing correlations cor If x is a data matrix cor x computes the correlation matrix of x If xis n bym cor x ism bym with 1 s down the diagonal and cor x i j the Pearson correlation between x i and x j Cmd gt w lt matrix vector 45 5 42 1 53 8 48 5 44 5 58 4 74 1 72 0 63
36. n advisory message The entire file will be scanned unless a is found Here is a listing of a file myfile dat containing 30 comma separated values of a variable x that might be read by vecread Data from trees 600 555 361 489 640 644 297 481 612 522 246 504 358 623 614 595 531 602 684 410 448 662 892 644 we 431 472 513 603e 30 10 10 20 3 This line will not be read The output of vecread is a REAL vector consisting of the values read 2 31 MacAnova Version 4 04 Cmd gt x lt vecread myfile dat x read from file myfile dat WARNI al 6 1 21 2 16 6 NG 600 M SSING 522 614 410 M SSING Note that and were read as M WARNI You can specify a different stopping character say vecread filename stop with the same character say that should be ignored you can use keyword phrase skip as an argument 60 24 0 6 662 10 Data on 555 504 892 20 30 36 35 64 This line will trees 1 489 8 623 4 640 614 431 nonnumeric character s SS 595 47 2 fio s 644 297 481 61 ignored on myfile dat 489 481 358 602 892 513 NG and nothing after the was read The NG message results from the non numeric first line instead of
37. nt boundaries 1 30 40 50 60 70 component counts 1 2 10 10 3 Some statistics books recommend that you include the left or lower end of a class interval in the interval instead of the right end You can follow this convention by including leftendin T as an argument Cmd gt bin x b leftendin T lower end of interval in class component boundaries 1 30 40 50 60 70 component counts 1 2 7 12 4 Now the new frequencies reflect the fact that the three 50 s are put in class 3 and the 60 in class 4 When you want equally spaced boundaries the usage bin x vector b0 width can simplify things bO is interpreted as a typical class limit and width is the desired class width Enough boundaries of the form b jxwidth are computed so that all the data are included Cmd gt bin x vector 30 10 gives the same result as bin x b component boundaries 1 30 40 50 60 70 component counts 1 2 10 10 3 Instead of vector 30 10 you could use vector 0 10 or even vector 100 10 and still get the same output Still easier and OK for quick summaries is bin x nclasses where nclasses is the number of classes wanted bin chooses nclasses 1 equally spaced class limits which include all the data No attempt is made to ensure the values are neat Cmd gt bin x 4 bin chooses boundaries component boundaries 1 32 15 41 5 50 85 60 2 69 55 component counts 1 4 8 10 3 Even
38. of the header and comment lines of the data set or macro Cmd gt info lt inforead macanova dat halddata qguiet T Cmd gt info see Sec 2 11 3 1 Hald data from A Hald Statistical Theory with Engineering Applications Wiley New York 1952 p 647 Col 1 X1 percent tricalcium aluminate Col 2 X2 percent tricalcium silicate Col 3 X3 percent tetracalcium alumino ferrite Col 4 X4 percent dicalcium silicate Col 5 Y cumulative heat evolved from cement hardening after 180 days calories gm inforead FileName Name quiet T notfoundok T behaves identically if the data set or macro is found However if it is not no message is printed and NULL is returned When used in a macro this feature allows special action if Name is not found Cmd gt appleinfo lt matread macanova dat appledata notfoundok T Cmd gt print appleinfo appleinfo NULL 2 11 6 HOME DATAPATHS and adddatapath When MacAnova is started up two CHARACTER variables HOME and DATAPATHS are created On all systems except Unix these are both initialized to the complete path name of the directory or folder where MacAnova is located On a Unix system HOME is initialized to the name of the user s home directory environmental variable HOME and DATAPATHS is initialized to the name of an installation dependent directory containing data files Both variables can be modified
39. or 27 3 20 8 32 0 28 8 18 9 28 1 22 97 824 32 41 B1 4 292 17 3143 pap 2 BF 3 bY 2 by 2 Cmd gt ary 2 2 1 1 1 1 31 24 2 1 7 1 29 1 3 1 1 S23 Cmd gt sort ary 2 2 same as sort ary 2 2 1 1 1 29 1 2 1 1 Le 3 155 31 4 It is also acceptable for x to be a structure Sec 2 8 16 whose non structure components are all REAL or all CHARACTER In that case each function returns a structure of the same form each of whose non structure components is the result of applying the function to the corresponding component of x Cmd gt sort structure dry vector 3 32 2 99 1 61 2 52 wet vector 3 60 4 21 3 636 component dry 1 1 61 22 2 99 3 32 component wet 1 3 6 3 636 4 21 You can modify the handling of ties in by rank rankits and halfnorm using keyword phrase ties method where method is one of average minimum or ignore which you can abbreviate as a b and i Suppose k elements in a vector column are tied that is they all have the same value and no other element has this value and suppose the ranks these elements would have if their values were very slightly changed so as to break the ties while preserving other ordering would ber r 1 r 2 r k 1 The following table describes the ranks computed by rank for the tied values for each of the three possible methods 2 47 MacAnova Version 4 04 average
40. ows computers you can use instead of in path names In fact if you want to use you have to use see Sec 2 5 2 11 5 inforead As exemplified in Sec 2 11 3 data sets on external files that are readable by mat read may have associated comment lines that give information about the data see also Sec 7 1 The same is true of macros on external files Sec 7 5 It can be helpful to save these comments in a CHARACTER variable so that they are instantly available for references You can do this using inforead which is used much the same as matread inforead FileName Name searches file FileName for a macro or data set with name 2 36 MacAnova Version 4 04 Name If found it reads the comments the lines starting with line and returns a CHARACTER variable containing these lines with the leading stripped off FileName and Name must be quoted strings or CHARACTER variables The actual contents of the data set or macro are ignored and there is no checking as to whether the header line is in correct format following the header es A inforead FileName does the same for the first data set or macro on the file assuming that line 1 is the header line In versions with windows Macintosh Windows Motif if FileName is the null string you will be able to select the file using a dialog box inforead FileName Name quiet T returns the comments but suppresses any echoing
41. readcols recognizes vecread keywords stop skip quiet silent and badvalue See Sec 2 11 1 for details 2 11 3 matread A single plain text file can contain several REAL LOGICAL or CHARACTER data sets provided each of them has a header consisting of at least one line of information specifying a name for the data set and its dimensions length for a vector numbers of rows and columns for a matrix Optional additional header lines can describe the data set and provide formatting information It can also contain NULL variables and structures See Sec 7 1 for details of the file format You use mat read to read from such a file Function matread normally requires two CHARACTER variables or quoted strings as arguments the file name and the data set name Thus Cmd gt x lt matread data txt treedata would search file data txt for a data set named t reedata read it and assign the data variable x If you omit the data set name mat read assumes that the first non empty line in the file is the first header line for the data set You can create files in the format readable by matread using matprint and matwrite see Sec 7 4 If the named file is not in the current default directory or folder mat read searches the directories 2 34 MacAnova Version 4 04 or folders in CHARACTER vector DATAPATHS See Sec 2 11 6 File MacAnova dat distributed with MacAnova contains the data in a form readable by matrea
42. rees of freedom cumstu x df Non central Student s t on df degrees of cumstu x df delta freedom noncentrality delta x on df degrees of freedom cumchi x df Non central y2 on df degrees of freedom cumchi x df noncen noncentrality noncen F on df1 and df2 degrees of freedom cumF x df1 df 2 Non central F on df1 and df2 degrees of cumF x df 1 d 2 noncen freedom noncentrality noncen Gamma with shape parameter alpha cumgamma x alpha Beta with parameters alpha and gamma cumbeta x alpha gamma Non central beta with parameters alpha and cumbeta x alpha gamma noncen gamma noncentrality noncen Binomial with n trials and success probability p cumbin x n p Poisson with mean lambda cumpoi x lambda 5 studrng x k df Studentized range for k groups df error degrees cu of freedom The noncentral versions of cumchi cumstu cumF and cumbeta are useful in computing the powers of tests that is the rejection probability when the null hypo thesis is not true In every case these CDF functions compute a lower tail probability that is the probability that the random variable is less than or equal to x All the parameters except noncentrality parameters must be positive Parameter noncen must be non negative for cumchi cumF and cumbeta Parameter n for cumbin and k for cumstudrng must be integers but other parameters including degrees of
43. roken If you use save every few minutes you minimize the danger of losing work To make this easier after the first use of save fileName simply save omitting fileName saves to the same file If the original saving was done by asciisave a plain text files will be written To save your workspace on a windowed version select Save Workspace on the File menu or press K on a Macintosh or Control K in the Windows or Motif versions The first time you do this you will be prompted for a save file Later uses of Save Workspace will save on the same file with no prompting See Sec 7 7 for information on partial saves and keywords all and delete 2 66
44. s optionally connected by lines lineplot x y Connected line plot of x versus y optionally augmented by impulses chplot x y sym supplied symbols points optionally connected by lines showplot Redisplay previously plotted graph possibly with changed labels or plotting limits Scatter plot or impulse plot of y versus x using user Makes parallel line plots of matrix x against row number Makes parallel line plots of matrix x against column number Impulses are vertical lines connecting a point with the x axis plot lineplot and chplot take at least two arguments x and y where x is a REAL vector and y is a REAL vector or matrix and plot each column of y against the values of x The length of x must normally match the number of rows in y but see below for a useful exception All plotting commands also recognize several optional keyword arguments which you can use to specify a title axis labels and minimum and maximum values to be plotted 2 15 1 plot plot x y does a simple scatter plot of y versus x If y has several columns a different symbol is used for each column Cmd gt xstuff lt vector 1 2 3 4 5 6 7 ystuff lt exp 5 xstuff 5 2 59 hhe tng MacAnova Version 4 04 Cmd gt plot xstuff ystuff simplest usage 1 Note that the axes are labelled with the variable names Mmhmhoatonk Cmd gt plot xstuff ystuff
45. s and tint and t2int compute one and two sample confidence intervals tval x where x is a REAL vector computes the usual one sample t statistic to test the null hypothesis Hp y 0 You can test a different null hypothesis say Ho u 100 by the statistic computed by tval x 100 When x and y are REAL vectors of the same length you can compute the paired t statistic to test the null hypothesis Hg p Hy by tval x y Cmd gt xl lt vector 3 2 5 4 6 8 6 4 3 7 3 Cmd gt vector tval x1 tval xl 5 test HO u 0 and HO p 5 1 8 0437 0 63088 1 sample t statistics Cmd gt x2 lt vector 4 2 5 6 7 9 7 4 3 8 5 tval x2 x1 1 3 6145 Paired t statistic When x and y are REAL vectors t2val x y computes a two sample t statistic to test the null hypothesis Ho u p or equivalently Ho 4 uy 0 To test a different null hypothesis say Ho uy u 10 use t2val x 10 y This use of t2val assumes the samples come from populations with equal variances and uses a pooled estimate of the standard deviation See below for the use of t2val when you cannot assume equal variances Cmd gt y lt vector 7 9 6 5 7 6 9 8 Cmd gt test HO ux py and H0 ux uy 3 Cmd gt vector t2val xl1 y t2val x1 3 y 1 3 0795 0 63278 2 sample t statistics See Sec 2 12 7 to see how to compute P values for these test statistics Functions tint and t2int compute one or two sample confidence inter
46. sav save workspace including vector x Workspace saved on file sessionl sav Cmd gt quit terminate the run Now suppose you start a new MacAnova session Cmd gt x At this point x is not defined UNDEFINED Cmd gt restore sessionl sav restore the save file Restoring workspace from file sessionl sav Workspace saved Thu Nov 30 11 52 58 1995 Cmd gt x variable x as we previously saved it is now defined 1 0 67374 0 13423 0 82378 0 89615 0 66544 The file created by save is in a binary format readable only by MacAnova running on the same type of computer It is useless on any other type of computer In contrast asciisave fileName saves your workspace in a form that can be read by MacAnova on any type of computer The file it writes is a plain text or ASCII file although it has an arcane format intended to be read only by a computer You can even safely send it via e mail with little danger of corruption since there are no non printable characters in the file Command restore can read either type of file although a file written by asciisave may take a little longer to restore The file created by asciisave is often larger than the one created by save although that need not be the case Another use for save and restore is when you are running MacAnova on an unstable computer subject to crashes or running a Unix version over the telephone using a modem where there is a chance the connection will be b
47. t case each component of describe x is itself a structure similar to x Cmd gt describe temperatures mean T var T component mean component Saturday 1 69 3 component Sunday 1 67 component Monday 1 66 786 2 39 MacAnova Version 4 04 component var component Saturday 1 14 233 component Sunday 1 3939 component Monday 1 19 412 Here temperatures is the structure used as an example in Sec 2 8 16 2 12 2 boxplot vboxplot stemleaf and hist These are intended to give a quick look at the distribution of a sample or samples of data boxplot x predefined macro vboxplot draw a Tukey box and whisker diagram often called a box plot of the data in x This is a graphical summary of the distribution of the values in x When x1 x2 xk are several variables boxplot x1 x2 xk draws parallel horizontal box plots a very useful way to compare the distributions of the different variables vboxplot x1 x2 xk If you prefer a plot box plots in which the boxes are oriented vertically use boxplot x vertical T or more simply vboxplot x1 x2 xk A box plot consists of a central box with a line drawn somewhere between the left lower and right upper ends whiskers extending off the ends of the box and separately plotted outliers The ends of the box are drawn at the lower and upper quartiles respectively and the middle line is the median The whiskers extend to the
48. trality parameter lambda F on df1 and df2 degrees of freedom invF p df1 df2 Gamma with shape parameter alpha invgamma p alpha Beta with parameters alpha and gamma invbeta p alpha beta Studentized range for k groups df error degrees invstudrng p k df of freedom In every case these inverse CDF functions compute a value xo such that P x lt x9 p That is they are the inverses of the corresponding CDF functions in Sec 2 12 7 The value of p must always be between 0 and 1 All the other parameters except k for invstudrng including degrees of freedom must be positive but need not be integers When pis a vector or matrix the inverse CDF is computed for each element of p producing a vector or matrix Similarly the parameters may be vectors or matrices The sizes and shapes of the arguments must match except that p or any of the parameters may always be scalars Cmd gt invnor vector 10 05 025 01 005 normal probability points 1 1 2816 1 6449 1 96 2 3263 2 5758 2 54 MacAnova Version 4 04 Cmd gt invstu 975 run 5 25 5 Student s t on 5 10 15 20 25 df 1 2 5706 2 2281 2 1314 2 086 2 0595 Cmd gt invF 95 5 run 5 25 5 F with df1 5 and df2 5 10 15 20 25 1 5 0503 3 3258 2 9013 2 7109 2 603 Cmd gt invstudrng 1l vector 1 05 01 001 k k n 1 1 3 65309 3 9328 4 7373 5 7199 The invstudrng output are the 10 5 1 and 0 1 critical values of the
49. tures would produce the same display of box plots A common way to create a structure for use with boxplot is to use split see Sec 9 1 1 In the following temps is a vector combining the data from the three components of temperatures and a day is a vector of integers coding Saturday as 1 Sunday as 2 and Monday as 3 Cmd gt print temps day format 2 0f see Sec 7 4 for use of format y 1 65 71 75 86 91 93 89 78 69 59 61 73 85 83 81 51 65 71 78 83 84 22 85 84 81 75 69 64 59 49 day OLE sls Salto Sil ak ES GL SE SHY SS A De Die DD ID OB gt Se OB Be Br 8 22 8 3 33 38 3 SB Cmd gt split temps day create structure splitting up temps by day component dayl 1 65 71 75 86 91 6 93 89 78 69 59 component day2 1 61 73 85 83 81 component day3 1 5I 65 pan 78 83 6 84 85 84 81 1D 11 69 64 59 49 Cmd gt boxplot split temps day This would produce the same horizontally oriented plot as before You can print a stem and leaf display of the data in a REAL vector or an by 1 matrix x by stemleaf x Cmd gt stemleaf temperaturesSMonday 4 9 SREE Sei 9 6 4 6 59 FALE Tal 58 8 1344 8a l5 S a a R E oE S OE 1 1 represents 11 Leaf digit unit 1 2 42 MacAnova Version 4 04 The numbers to the left of are the stems are represent the first digit sometimes 2 digits of each number The numbers to the right of are l
50. vals based on Student s t In a sense they are complementary to tval and t2val Their output is a REAL vector of length 2 containing the lower and upper ends of the confidence interval When x is a REAL vector and coverage is a REAL scalar tint x coverage returns a one sample confidence interval for the mean u with confidence level coverage For obvious reasons 0 lt coverage lt 1 For instance you can compute a 95 confidence interval by tint x 95 When x and y are two REAL vectors of data t2int x y coverage computes a two sample confidence interval for the difference u u of their means where coverage is as for tint Like t2val function t2int assumes equal variances and uses a 2 50 MacAnova Version 4 04 pooled estimate of the common variance in computing the standard error of the difference of means See below for usage not requiring equal variances Cmd gt tint x1 95 1 3 23 02 il 5 9207 95 confidence interval for yp Cmd gt t2int xl1 y 95 1 4 1936 0 78364 95 confidence interval for px py Both t2val and t2int assume by default that o o o where oj and o are variances of the two populations being compared Because of this assumption they use 1 1 n 1 s n Ds3 the estimated standard error of X S where s CIT a ne vere n on n n 2 is the pooled estimate of o2 based on the two sample variances s and s and has n n
51. var2 as 2 min cumF varl var2 df1 df 2 l cumF varl var2 df1 df2 Cmd gt vector cumnor 1 96 cumstu 2 1 10 normal Student s t 10 1 0979 0 031039 Cmd gt 2 1 cumstu abs 2 1 10 2 tail t P value 1 0 062077 Cmd gt vector l cumchi 34 25 1 cumF 4 5 2 10 chi squared and F 1 0 10791 0 040386 Cmd gt l cumstu 2 1 vector 10 20 30 40 50 5 d f s at once 1 0 031039 0 024309 0 022121 0 021041 0 020398 Cmd gt xbar lt vector 150 91 150 77 159 51 156 43 165 54 enter means Cmd gt spooled lt 15 974 enter pooled var from k 5 groups of n 20 Cmd gt n lt 20 k lt 5 Cmd gt studrng lt max xbar min xbar spooled sqrt n studrng 1 4 1351 Studentized range statistic Cmd gt 1 cumstudrng studrng k k n 1 compute P value 1 0 034262 Significant at 5 level You can compute binomial and Poisson probabilities of the form P X x by taking differences of cumulative probabilities Cmd gt p lt 3 n lt 9 p amp lt cumbin 8 n p cumbin 7 n p pe 1 0 00041334 P X 8 Cmd gt n 1 p 8 1 p n 8 n 1 is binomial coefficient n 1 1 0 00041334 Confirmation of value You can even compute P X 0 P X n in a single line Cmd gt cdf x cumbin run 0 n n p cdf vector 0 cdf n 1 1 0 040354 0 15565 0 26683 0 26683 0 17153 6 0 073514 0 021004 0 0038579 0 00041334 1 9683e 05 The studentized range is R s where R Xmax
52. words title and xlab Examples might be Cmd gt boxplot temperatures title Saturday through Monday Temperatures Xlab Degrees Fahrenheit Cmd gt hist temperaturesSMonday vector 40 50 60 70 80 85 90 title Monday temperatures xlab Degrees Fahrenheit Normally hist considers an upper or right class limit to be in a class and the lower or left class limit to be in the next lower class If you include leftendin T as an argument to hist the lower limit is in the class and the upper limit is in the next higher class This corresponds to the convention used in some statistics text books 2 12 3 sort rank grade rankits and halfnorm These functions all have to do with the ordering of data values and expect a single vector or matrix argument plus an optional keyword phrase down T All work with REAL data sort rank and 2 44 MacAnova Version 4 04 grade also work with CHARACTER data For CHARACTER data ordering is in alphabetical order using the ASCII collating sequence in which most punctuation and all numerals sort ahead of upper case letters which sort ahead of lower case letters A space sorts ahead of all printable characters Here is the explicit ordering starting with space Sorting order of characters I S2E 0123456789 lt gt ABCDEFGHIJKLMNOPORSTUVWXYZ _ abcdefghijkimnopgrstuvwxyz A null string sorts before any other string
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