Nonparametrics on Minitab Version 1

Transcription

1 θ Nonparametrics on Minitab Version 1? ρπθ σχωµ σχωµ µρπ ρ πθ σ χω Stat By Julian Visch & Irene Hudson Department of Mathematics and Statistics University of Canterbury

2 Nonparametrics on Minitab Based on Tony Davidson s A Students Introduction to Minitab created on June 21, 2000 using AMS-LaTeX

3 Contents 1 Introduction Terminology Used General Advice Where to find the Help Desk Getting Started Finding the Computer Rooms Getting into the Buildings Finding a Computer Using Minitab Getting into Minitab The Minitab Window The Minitab Environment(as detailed in the help menu) Minitab Commands The Introductory Minitab Session Introduction Reading a Command Information Command Print Command Name Command Describe Command Histogram Command Table Command Count and Tally Commands Copy Command LET Command. Arithmetic in MINITAB Data Entry Saving your data Retrieving Data Nonparametrics Stest Sinterval Wtest Winterval Mann-Whitney Kruskal Wallis Friedman Walsh or Pairwise Averages Pairwise Differences Wslope Rregress ii

4 1 Introduction A Students Introduction to MINITAB This booklet has been compiled to introduce students to using the Minitab package on the University s undergraduate computer system for statistical computations. 1.1 Terminology Used <key> means press the key shown in bold inside the <>for instance: <Enter> means press the Enter key. Text to be typed is shown in bold for instance: 72<Enter> means type 72 and then press the Enter key. Some commands require a key to be held down while another is pressed, this is shown with + between the keys e.g. <Ctrl>+<z> means hold down the Ctrl key and press z 1. Sometimes user supplied input is needed; this will be shown by italics. Note: While following the instructions, read ahead. This will help with your understanding and guard against you doing the next step wrongly, and getting confused. 1.2 General Advice It is essential that you understand clearly what each statistical procedure does. Writing down the result of the procedures is one way to achieve this. In future work, you will have to decide for yourself the following: What information do I require (from a MINITAB worksheet for a particular purpose)? How will I obtain this information? (i.e. what MINITAB commands should be used to obtain the information required?) At any time, personal help may be obtained from the Help Desk on Level 2 of the Computer Services Centre (phone 6060). Questions to the Help Desk should be of a general nature. For questions specific to this course, see your tutor or contact the Statistics secretary in order to be pointed in the right direction. 1 Usually in such cases upper or lower case is acceptable 1

5 1.2.1 Where to find the Help Desk 2

6 2 Getting Started 2.1 Finding the Computer Rooms There are several clusters of IBM Personal Computers (PCs) that will be available for your use. One cluster is called The Vault which consists of 5 individual rooms labelled Vault 1-5, with each room containing 23 PCs and a laser printer and is located in the basement of the Commerce building. Another cluster is called The Cave which is in the Engineering Library Extension Building on the north side of campus which consists of 48 PCs. A third cluster is called The Loft which has individual PCs and MacIntosh s available for individual use at all times and is located on the 5th floor of the Central Library (James Hight). And lastly there is the cluster called The Crypt which consists of 2 labs; Crypt1 and Crypt2. which is in the new Maths and Computer Science building. See below for a map showing these locations on campus. Math/Comp Building Cave Information Technology Services Crypt Loft Vault 3

7 2.2 Getting into the Buildings Access to either computer facility is by plastic card which you should have obtained when you enrolled. If you have not received a card or the card does not operate both of the doors, see the people at Registry. Please ensure that you have your card and that it works before the commencement of the Computer Familiarisation Class. To enter the building just swipe the card through the thin slot, then type in your security number, then the in button. You should then be able to just pull open the door and go in. 2.3 Finding a Computer When entering the James Hight, you will need to enter through the main doors opposite registry, take the lift(or the stairs) up to the fifth floor. In the Cave the computers are on the right, at the end of the short corridor. Vault 1 is in the Commerce building basement to left when exiting the lifts. Vaults 2-4 are located along the same side as vault 1, and vault 5 is located near to vault 4 but towards the center of the basement. The loft is upstair in the James Hight, and Crypt1 is in the Math/Comp building down the central stairs and then to your left. 4

8 3 Using Minitab 3.1 Getting into Minitab To get into Minitab, click the left mouse button on the Start Menu, move the mouse 2 to programs, then along to Math & Stats and then go along to Minitab and select. 3.2 The Minitab Window At this stage you should be in Minitab with a screen something similar to that shown below. Input and Output is viewed here in the Session Window Data can be entered into the columns of the Data window Titles can be inserted along this row only 2 You do not need to grip the mouse hard to do this 5

9 3.3 The Minitab Environment(as detailed in the help menu) 1. A Worksheet that contains your data. This includes constants, vectors(columns) and matrices. 2. A Data window that shows columns of data as shown on the previous page. In the Data window you can enter columns of data into the worksheet name, resize, and format columns move quickly to different cell locations cut, copy, or paste cells to and from the Clipboard Although the Data window has rows and columns, it is not a spreadsheet like Microsoft Excel or Lotus In Minitab, cells contain values that you type or generate with commands. Cells do not contain formulas that update based on other cells. For example, if you want column C3 to equal the values in C1 plus the values in C2, you would use the calculator (Calc > Calculator) to generate the values for C3. If you change the values in C1, C3 does not change until you use the calculator again or use some other command to change C3s contents. 3. Menus to issue commands for statistical analysis, data manipulation, and data transformation. Menu items can directly execute a command, or open a dialog box. These menus lie at the top of your Minitab window. As frequent menu use contributes to R.S.I. or O.O.S. and tends to be slower we recommend that you wherever possible stick to using the session window. 4. A Session window where you can type in your commands and displays your results as shown on the previous window. 5. Graph windows for high-resolution graphs. See the help facility (help menu) for details. 6. An Info window that displays a summary of your worksheet. See the help facility (help menu) for details. 7. A History window that lists commands you have used in your session. You can re-execute commands by copying them from the History window and pasting them into the Command Line Editor. See the help facility (help menu) for details. 6

10 8. Session commands are alternatives to menu commands that you can type in the Session window To type commands in the session window, you must first select Enable Command Language from the Editor menu (note not the Edit menu). To undo the process you must select Disable Command Language from the Editor Menu. Command Line Editor which allows you to quickly edit and reexecute session commands. The Command Line editor can be found on the Edit menu or can be activated using <Ctrl>+L. You can intersperse menu commands and session commands throughout your session if you wish. 9. Context-sensitive Help for dialog boxes, Session window commands, and overview information. 10. A complete macro language that lets you automate repetitive tasks, extend Minitab s functionality, or even design your own session commands. See the help facility (help menu) for details. 3.4 Minitab Commands 1. MINITAB commands may make reference to a column using the number of the column or its name (if it has been given one). If a name is used, it must be put between single quotes in the command, e.g. histogram item 2. The first four letters of any command are sufficient to identify it, e.g. hist item would have done in the example above. 3. If you make a mistake in typing a command, simply re-type it. This overwrites the previous statement. 4. Some commands have subcommands. To invoke a subcommand, type a semicolon at the end of the main command. This results in the prompt SUBC> appearing; then type in the subcommand. A subcommand must conclude with either a semicolon (if there is another subcommand you wish to issue) or a full stop (if there is not another subcommand). 7

11 4 The Introductory Minitab Session 4.1 Introduction I will assume that you have already logged onto the computer and have clicked onto the Minitab icon, and have the Minitab window open before you. Now select Enable Command Language from the Editor (not the edit Menu) menu and select Save Preferences from the Edit menu. This will allow you to enter your commands in the upper half of your Minitab Window. 4.2 Reading a Command You will now need to read in the inventory file. To do this, type Note: read k:parts.dat c1-c6 1. There is a single (forward) quote before the K: and after the filename. (The single forward quote is on the same key of the keyboard as the double quote.) 2. K: means that the file is being read from drive K (The class drive, note the class drive may differ for you). 3. The filename is parts.dat. 4. The file is a datafile, and hence the extension.dat as part of the filename. 5. All six columns of the file are being read. We could have read any columns we wanted, e.g. c1 or c1 c3 or c1 c3-c6 You may need to wait while the file is being read. You will be advised onscreen when this has been done, and the Minitab prompt will appear again. 4.3 Information Command Type info to obtain a summary of your worksheet. What does the info command give? 8

12 4.4 Print Command To see a portion of your worksheet, type PRINT C1-C6 (or just PRIN C1-C6). Note the prompt at the bottom of the first screen will require a response, Y for yes or N for no. Pressing the Return key has the same effect as Y. Type PRIN C1 to see column 1 only. Note the difference between printing a single column and printing more than one column. 4.5 Name Command You can name each column as follows. Type name c1 partno. You can use any name instead of partno, of course, but it is clearly advantageous to use a name indicative of what it represents. A name may have up to eight characters, may not start or end with a blank, and cannot contain the forward quote ( ) or hash (#). Further, the name must be put between single quotes. Several columns can be named at once, e.g. name c1 c2. Once named, a column may be referred to in a Minitab command by its name, but the name must be in single quotes in the command. Now you can name the other columns if you wish. 4.6 Describe Command 1. Type desc c4. What information does the describe command give? You can type help desc to find out the meaning of any of the things appearing with the desc command. 2. Now type desc c4; by c6. What is the effect of the subcommand? 9

13 4.7 Histogram Command 1. Type hist c2. 2. Type hist c2 c6 In particular, note the heading MIDPOINT and compare the differences between the two histograms 3. Now type hist c2; by c6. What is the effect of the by subcommand? 4. Now try the start and increment subcommands by typing HIST C2; STAR 15. What is the effect here? HIST C2; INCR 10. What is the effect here? 5. Investigate the subcommands with other values if you wish. 6. The STEM-AND-LEAF command has the subcommands INCREMENT and BY. The BOXPLOT command has the subcommands START, IN- CREMENT and BY. You may wish to investigate these commands and their subcommands. 10

14 4.8 Table Command 1. Type TABLE C5 What does this command give? 2. Type TABLE C5 C6 What does this command give? 3. Type TABLE C6 C5 What does this command give? Compare with the previous case. 4. Type TABLE C6 C5; ROWP. What does ROWP stand for? What output is produced? 5. Type TABLE C6 C5; TOTP. 4.9 Count and Tally Commands 1. Type COUNT C5 What output is produced? 2. Type TALLY C5 C6 What output is produced -i e.what does the TALLY command do? 11

15 3. Some of the subcommands TALLY has are COUNT CUMCNT (cumulative count) PERCENT CUMPCT (cumulative percent) Investigate these subcommands to find out what output they give Copy Command 1. Type COPY C6 C7 (the full command is COPY C6 INTO C7) Type PRIN C1-C7 to see the affect of this. What has happened? 2. Type COPY C1-C4 C7-C10 (or COPY C1-C4 INTO C7-C10) Describe What happens here. (In particular, note what has now happened to C7, which was created in (1) above.) 3. Now delete the extra columns you have created by typing ERASE C7- C10 (or just ERAS C7-C10). Suppose we wanted the part numbers of those items which are supplied from source 3. We can find this by typing COPY C1 C7; USE C6=3. Print C1, C6 and C7 to see the effect of this. Investigate the following to see its affect. COPY C1 C7; OMIT C6=1:2. 4. Suppose you wanted to know the part numbers of those items which were ordered in month 5. Write the appropriate MINITAB commands to find this, then try the commands. 5. Write the appropriate MINITAB commands to find how many parts cost more than $10.50, then try the commands. Note: The USE subcommand can be combined with the OMIT subcommand, e.g. COPY C5 C8; USE C5=0:4; OMIT C4=3. 12

16 However, USE and OMIT subcommands cannot be combined together as follows: COPY C5 C8; USE C5=0:4; USE C4=2. nor COPY C5 C8; OMIT C5=3; OMIT C4= LET Command. Arithmetic in MINITAB 1. Suppose the prices in C2 did not include G.S.T., and that we wished to calculate this and show the price including G.S.T. We use the following commands (for simplicity, we have taken the G.S.T. rate to be 10 LET C7=0.1*C2 (the values in C2 are multiplied by 0.1, and entered in C7) LET C8=C2+C7 (the values in C2 and C7 are added, and entered in C8). Try these, and then print C2, C7 and C8 to see the effect. (The prices in C7 and C8 will not have been rounded to the nearest cent.) Note that if we simply wanted the price including G.S.T. and didn t want the G.S.T. itself, we could have used the command LET C8=0. 1 *C2+C2 or simply LET C8=l.l*C2. The LET command is used for arithmetic operations, and uses the following where needed. + for addition - for subtraction for multiplication / for division * for exponentiation (raising to a power). 2. The LET command may also be used as follows: LET K1=MEAN(C2) which will assign the mean price (column 2) to a constant, K1. Typing PRIN K1 will allow you to see the result. Similarly, the MEDIAN or STDEV of a column could be round (they should be assigned to different constants, say K2 and K3). These constants could be used as follows: LET C9=(C2-Kl)/K3. 13

17 4.12 Data Entry You can enter data into Minitab as follows, you can either enter it directly into the cells or you can use the session window. e.g. 1. Type set c end e.g. 2. Type set c2 4(123) end What happened to the data in c2? e.g. 3. Type set c3 (1 2 3)4 end What happened to the data in c3? Enter in the following data into c What did you type? 14

18 4.13 Saving your data Command Syntax SAVE [in file in "filename" or K] After using SAVE, the file will contain all data in the worksheet, all stored constants, matrices, column names, and missing value information. After the file is RETRIEVED, the worksheet, stored constants, matrices, and column names will be exactly as when they were SAVED. You may specify the filename as either the name of the file in double quotes, or a stored text constant. A SAVED worksheet can be used only by Minitab s RETRIEVE command. You cannot edit it with an editor or even look at it. Unless you use the subcommand PORTABLE, you can RETRIEVE it only on the same type of computer on which it was SAVED. For most applications, however, SAVED worksheets are the most efficient and convenient way to store data for use in Minitab. If a file name is not given, the default name MINITAB.MTW is used. If you use RETRIEVE without a file name, this default file is retrieved. The default file is useful for saving temporary copies of your worksheet, throughout your session, as a backup in case you accidentally destroy the worksheet. The default file extension for SAVE without any subcommands is MTW. By default, if you SAVE "filename" when the file already exists, Minitab asks you whether or not you want to replace the file before proceeding. If you SAVE without a file name or you are in BATCH mode, Minitab automatically replaces the file. You can use the subcommands REPLACE and NOREPLACE to override Minitab s default behavior Retrieving Data Command Syntax RETRIEVE [file in "filename" or K] Retrieves a saved worksheet from the specified file into the current worksheet. You may specify the filename as either the name of the file in double quotes, or a stored text constant. Following this command, the worksheet will contain the same numbers, column names, stored constants, and matrices as when the command SAVE was last used to save them all. If there is any data in the current worksheet, RETRIEVE erases that data and replaces it with the specified saved worksheet data. To add data to the current worksheet without replacing it, use READ or INSERT. If you omit the file name, Minitab looks for a file in your current directory named MINITAB.MTW. The menu command File > Open Worksheet also opens Minitab saved worksheets and Lotus files (and many other types of files as well). It also provides several useful options not available with RETRIEVE. See File > Open Worksheet for details. For information on open data sets that come with Minitab, see Retrieving Sample Data Sets. 15

19 5 Nonparametrics 5.1 Stest Command Syntax STEST sign test [median = K] on C...C ALTERNATIVE = K. Example: Suppose we take a sample of 29 observations from a population with median = 115. First let s test Ho: median = 115 H1: median > Statistical Rationale Let X be the number of observations over 115. X has a binomial distribution with n = 29 and p = 0.5. Here X = 17. The probability of getting 17 or even more observations over 115 is , which is the p-value for this test. To do this analysis using Minitab you will first have to read the data set stest.dat, then type MTB> stest 115 C1; SUBC> Alternative 1. Note: 115 is the median you wished to test and specifying the alternative to be 1 informs Minitab that the alternative is median > 115. If one want the alternative to be < 115 then replace 1 with -1. For a two sided test omit the alternative subcommand. Exercise 1. Now suppose we wish to test Ho: median = 115 H1: median < 115 What is X in this case? What is the p-value in this case? Exercise 2. Now suppose we wish to test Ho: median = 115 H1: median = / 115 What is X in this case? What is the p-value in this case? 16

20 5.2 Sinterval Command Syntax SINTERVAL sign CI [K% confidence] on C...C Calculates a sign confidence interval, separately for each column. You can specify a confidence level on the command line. For example, if you enter the command SINTERVAL.90 C1 Minitab calculates 90% confidence intervals. If you do not specify a confidence level, SINTERVAL gives a 95% confidence interval. Example: Suppose we take a sample of 29 observations from a population and wish to calculate the sinterval for its median To do this analysis using Minitab you will first have to read the data set stest.dat if you haven t already done so, then type MTB> sinterval C1. Statistical Rationale Minitab calculates three intervals. The first gives the achievable confidence just below K, and the third the achievable confidence just above K. Only rarely can you achieve confidence K using the standard procedure. The middle confidence interval is found by a nonlinear interpolation procedure, and gives an interval with approximate confidence K. The three confidence intervals are found as follows: Let M be the true, unknown median. Suppose we take a sample of n observations. Let X be the number of observations which are less than M. X has a binomial distribution with parameters n and p = 0.5. To calculate a sign confidence interval, first rank the n observations. The interval that goes from the dth smallest observation to the dth largest observation has confidence 1-2P (X < d). The value of d is given under POSITION on the output, for the first and third confidence interval. The middle confidence interval is found by a nonlinear interpolation formula (denoted by NLI in the POSITION column). This method has the following properties: (a) the actual confidence level is between the confidence levels for the bounding intervals, (b) the interpolation is a very good approximation for a wide variety of symmetric distributions including the normal distribution, the Cauchy distribution, and the uniform distribution, and (c) examples of nonsymmetric distributions studied show fairly reasonable results, always much more accurate than linear interpolation. Exercise 1. What is meant by the three achieved confidences, and how are they calculated? Exercise 2. What does NLI stand for? 17

21 5.3 Wtest Command Syntax WTEST Wilcoxon one-sample rank test [of median = K] on C...C ALTERNATIVE = K. Performs a one-sample Wilcoxon signed-rank test of the median. If you do not specify a hypothesized median, WTEST compares the sample median to 0. Example: First let s test Ho: median = 77 H1: median 77 With C1: After typing in the data, type wtest 77 c1. Statistical Rationale Minitab first eliminates any observations equal to the hypothesized median. The number of observations remaining is printed on the output as N FOR TEST. Then the pairwise (Walsh) averages, (Yi + Yj)/2 for i < j, are formed. The Wilcoxon statistic is the number of Walsh averages exceeding the hypothesized median, plus one half the number of Walsh averages equal to the hypothesized median. This statistic is approximately normal. Under Ho, it has mean N (N + 1)/4, where N is the number of observations for the test. The attained significance level, or p-value, is calculated using a normal approximation with a continuity correction. An algebraically equivalent form of the test is based on ranks. Subtract the hypothesized median from each observation, discard any zeros, and rank the absolute values of these differences. The number of differences is N FOR TEST. If two or more absolute differences are tied, assign the average rank to each. The Wilcoxon statistic is the sum of ranks corresponding to positive differences. The Wilcoxon point estimate of the population median is the median of the Walsh averages. Minitab obtains the test statistic and point estimate of the population median using an algorithm based on Johnson and Mizoguchi. Exercise 1. Would you accept or reject the null hypothesis? Why? Exercise 2. Now suppose we wish to test Ho: median = 77 H1: median > 77 What do you need to type in this case? Exercise 3. What is the p-value in this case? And what is your conclusion? 18

22 5.4 Winterval Command Syntax WINTERVAL Wilcoxon CI [K% confidence] on C...C Calculates a one-sample Wilcoxon confidence interval, separately for each column. You can specify a confidence level on the command line. For example, if you enter the command WINTERVAL.90 C1 Minitab calculates a 90% confidence interval. If you do not specify a confidence level, WINTERVAL gives a 95% confidence interval. Example: To calculate a 1-Sample Wilcoxon confidence interval. With C1: Type wint c1. Statistical Rationale The confidence interval is essentially the set of values, d, for which the test of Ho: median = d is not rejected in favor of H1: median not equal to d, using a = 1 - (percent confidence)/100 for Example of 1-Sample Wilcoxon Confidence Interval. You do not reject Ho as the estimated median, 77.5, is contained in the confidence interval Because of the discreteness of the Wilcoxon test statistic, it will seldom be possible to achieve the specified confidence. Minitab prints the closest value, which is computed using a normal approximation with continuity correction. Exercise 1. What is the level of confidence achieved? Exercise 2. What is the confidence interval? Exercise 3. Given the data C2: Find a 80% confidence interval. a. What is the level of confidence achieved? b. What is the confidence interval? 19

23 5.5 Mann-Whitney Command Syntax MANN-WHITNEY two-sample rank test with [K% confidence] on CC ALTERNATIVE = K Does a two-sample rank test (often called the Mann-Whitney test, or the two-sample Wilcoxon rank sum test) for the difference between two population medians, and calculates the corresponding point estimate and confidence interval. You can specify a confidence level on the command line. If you do not specify a confidence level, MANN-WHITNEY gives a 95% confidence interval. Example: Using the data below we can calculate a Mann-Whitney confidence interval and test. C1: C2: MTB> mann-whitney C1 C2. Statistical Rationale First, the two samples are ranked together, with the smallest observation given rank 1, the second smallest, rank 2, etc. If two or more observations are tied, the average rank is assigned to each. Then, the sum of the ranks of the first sample is calculated. This sum is the test statistic, W. A small value of W indicates that M1 is smaller than M2; a large value indicates that M2 is smaller than M1, where M1 and M2 are the population medians. Minitab obtains the attained significance level of the test using a normal approximation with a continuity correction factor. If there are ties in the data, the significance level adjusted for ties is also printed. The unadjusted significance level is conservative if ties are present; the adjusted significance level is usually closer to the correct values, but is not always conservative. The point estimate is the median of all the pairwise differences between observations in the first sample and observations in the second sample. The confidence interval is the set of values d for which the test of Ho: M1 - M2 = d versus H1: M1 not equal to M2 is not rejected, at a = 1 - (percent confidence)/100. Exercise 1. What is being tested? i.e. What is the hypothesis test? Exercise 2. What was the conclusion? Exercise 3. What results do you get if you conduct a 85% confidence test. 20

24 5.6 Kruskal Wallis Command Syntax KRUSKAL-WALLIS test for data in C, levels in C The factor column may be numeric or text, and may contain any value. The levels do not need to be in any specific order. Example: Perform a Kruskal Wallis test for the following data C1: C2: This test is a generalization of the procedure used by MANN-WHITNEY, and offers a nonparametric alternative to the usual one-way analysis of variance. The test assumes that the data arise as k independent random samples from continuous distributions, all having the same shape. The null hypothesis of no differences among the k populations is tested against the alternative of at least one difference. The factor column may be numeric or text, and may contain any value. The levels do not need to be in any special order. 21

25 Statistical Rationale First the combined samples are ranked. If two or more observations are tied, the average rank is assigned to each. The test statistic is H = 12 n j [ R j R] 2 N(N +1) where n j is the number of observations in group j, N is the total sample size, R j is the average of the ranks in group j, and R is the average of all the ranks. Under the null hypothesis, the distribution of H can be approximated by a chi-squared distribution with k - 1 degrees of freedom. The approximation is reasonably accurate if no group has fewer than five observations. Large values of H suggest that there are some differences in location among the k populations. Some authors (e.g., Lehmann) suggest adjusting H when there are ties in the data. Suppose there are J distinct values among the N observations and, for the jth distinct value, there are dj tied observations (dj = 1 if there are no ties). Then H(adj) = H 1 [ (d 3 j d j)/(n 3 N)] When there are no ties, H(adj) = H. Under the null hypothesis, the distribution of H(adj) is also approximately a chi-squared with k - 1 degrees of freedom. For small samples, we suggest the use of exact tables (e.g., Hollander and Wolfe). Minitab prints H(adj) if there are ties. The following z-value is printed for each group. For group i, z j = R j (N +1)/2 (N +1)(N/nj 1)/12 Under the null hypothesis, z j is approximately normal with m = 0 and s = 1. The value of z j indicates how the mean rank, R j, for group j differs from the mean rank, R-bar, for all N observations. 22

26 Exercise 1. What is being tested? i.e. What is the hypothesis test? Exercise 2. What was the test statistic in this case? Exercise 3. What was the conclusion? 5.7 Friedman Command Syntax FRIEDMAN data in C, treatment in C, blocks in C [put residuals in C [fits in C]] Does a nonparametric analysis of a randomized block experiment, and thus provides an alternative to the command TWOWAY. Randomized block experiments are a generalization of paired experiments, and FRIEDMAN is a generalization of the paired sign test. FRIEDMAN tests the null hypothesis that treatment has no effect. Additivity is not required for the test, but is for the estimate of the treatment effects. The first column listed on the command line contains the response data; the second column, treatment levels; and the third column, blocks. The treatment and blocks columns may be numeric or text, and may contain any values. The levels do not need to be in any special order. Optionally, you can store the residuals by adding a fourth column; fits, or group medians, by giving a fifth column. This command requires exactly one observation per cell; missing data are not allowed. Minitab prints the test statistic, which has an approximately Chi-square distribution, and the associated degrees of freedom (number of treatments - 1). If there are ties within one or more blocks, the average rank is used, and a test statistic corrected for ties is also printed. If there are many ties, the uncorrected test statistic is conservative; the corrected version is usually closer, but may be either conservative or liberal. FRIEDMAN displays an estimated median for each treatment level. The estimated median is the grand median plus the treatment effect. Example: Using the data below we can perform a Friedman Test for a randomised block design. C1: C2: C3: MTB> friedman C3 C1 C2. 23

27 Statistical Rationale To calculate the test statistic S, first rank the data, separately within each block. Then sum the ranks for each treatment. The test statistic is a constant times [(Rj R) 2 ], where R j is the rank sum of ranks for treatment j, and R is the average of the R j s See standard nonparametric texts, for details on computing S adjusted for ties. To calculate the treatment effects (Doksum method), first find the median difference between pairs of treatment. For the data above, the pairwise differences for treatment 1 minus treatment 2 are = -0.4, = 0, (-0.22) = 0.45, and = 0. The median of these is 0. Doing this for the other two pairs gives -0.4 for treatment 1 minus treatment 3, and -0.2 for treatment 2 minus treatment 3. The effect for each treatment is the average of the median differences of that treatment with all other treatments (including itself). For the data in Example of Friedman Test, effect(2) = [median (2-1) + median (2-2) + median (2-3)]/3 = ( )/3 = Similarly, effect(1) = and effect(3) = Adjust each observation by subtracting the appropriate treatment effect from the observation. Adjusted block medians are simply the block medians of the data adjusted for treatment effect. The grand median is the median of these adjusted block medians. The estimated median for each treatment level is the treatment effect plus the grand median. (Note: The average of the treatment medians is the grand median.) Residual = (observation adjusted for treatment effect) - (adjusted block median). Fit = (treatment effect) + (adjusted block median) = (observation) - (residual). Exercise 1. Which of C1-C3 is the response variable? Exercise 2. How was one able to type in the data for C1 quickly? Exercise 3. What is being tested? i.e. What is the hypothesis test? Exercise 4. What was the test statistic? Exercise 5. What was the conclusion? 24

28 5.8 Walsh or Pairwise Averages Command Syntax WALSH averages for C, put into C [put indices into C C] Calculates the average of all possible pairs of values, including each value with itself. Let x1,x2,...,xn be the observations. The Walsh average, (xi + xj) / 2, has indices i and j. If you specify index columns, the value of i is put in the first column and j in the second column. If you have n observations, there will be n (n + 1) / 2 Walsh averages. This command is useful for nonparametric tests and confidence intervals. Example C1: MTB > WALSH C1 C2 C3 C4 This command does not produce output in the session window. results, look in the data window. To see the Exercise 1. How would one display the data in the session window? Exercise 2. How was C2 calculated? Exercise 3. What is C3 and C4? Exercise 4. How would you go about naming the columns? 5.9 Pairwise Differences Command Syntax WDIFF for C and C, put into C [put indices into C and C] Computes all possible differences between pairs of elements from two columns by subtracting a value in the second column from the corresponding value in the first column. Let x1,x2,...,xn be the values in the first column, and y1,y2,...,ym be the values in the second. WDIFF finds all the differences, (xi - yj). If you specify index columns, the value of i is put in the first column and j in the second column. These differences are useful for nonparametric tests and confidence intervals. For example, the point estimate given by MANN-WHITNEY can be computed as the median of the differences. 25

29 5.10 Wslope Command Syntax WSLOPE y in C, x in C, put slopes into C [put row indices into C C] This command is useful in finding robust estimates of the slope of a line through the data. Each row of the y-x columns defines a point in the plane. WSLOPE computes the slope between every pair of points and stores the slopes in the third column given on the command line. If you specify index columns, the two row numbers used to compute the slope are put in the corresponding row of these two columns. If any observations are missing or if the slope cannot be defined (e.g., the slope of a line parallel to the y-axis), the slope is computed as missing. If there are n rows in the input columns, there will be n (n - 1) / 2 slopes in the output column. Example C1:3526 C2: MTB > wslope c1 c2 c3 c4 c5 Exercise 1. What is stored in C4 and C5? Exercise 2. How are the elements of C3 calculated? Give 2 examples of their calculations Rregress Command Syntax RREGRESS y in C on K predictors in C...C NORMAL scores WINSORIZED Wilcoxon scores with fraction K sign scores WILCOXON scores LEHMANN scale estimate [with t = K] WINDOW scale estimate [with shape = K] COEFFICIENTS in C FITS in C PSEUDO observations in C RESIDUALS in C NOEQUATION HYPOTHESIS matrices M...M QFORM ITERATIONS K STARTING values in C STEPINFO [K] TOLERANCE K for dispersion Note: RREGRESS is an experimental command. 26

30 Performs rank regression. The method for estimating the regression coefficients is an extension of the Mann-Whitney-Wilcoxon procedure for the twosample problem. RREGRESS offers a robust, asymptotically distribution-free alternative to the usual least-squares analysis. The regression coefficients are found by minimizing a measure of the dispersion of the residuals. Click on the help menu and select search for help on, select find, find rregress and click on Display. You will find the same details as given here in this booklet. You will notice that some items have been underlined. Exercise 1. What is the general term used to describe such items? Exercise 2. Click on Normal, what happened? Exercise 3. rregress y in C on K predictors in C..C means what? Give an example of its use. Exercise 4. If one wanted the printing of the regression equation and the table of coeficients and their standard error to be suppressed, what would you type? 27