Nonparametric Methods

Transcription

1 1 Excel Manual Nonparametric Methods Chapter 15 In this chapter, Excel is used more as an organizer; the actual statistics will performed by hand. Excel is used to do the simple task of ranking data, determining positive or negative differences, or doing both in the same problem. These tasks are not difficult to do by hand, but if the data set is large, it s very easy to make a simple mistake. Excel automates the process. Tests About the Median Difference Between Paired Data Excel can be used to illustrate how the sign of the difference may be found and counted for use in various tests. =SIGN(value) returns a 1 if the value is positive, -1 if it is negative and 0 if the value is 0. The following example uses the data from Example 15-7 on page 646 in the online-only portion of the text, from the results of a speed writing course for secretaries: Enter the data for Secretary, before speed and after speed in columns A, B and C, respectively. D1: Difference E1: Sign D2: =B2-B3 to calculate the difference. E2: =SIGN(D2) will return a 1 for positive, -1 for negative and 0 for 0. Select D2 and E2 and autofill these cells down to the end of the data (i.e. D9:E9). The following illustration depicts the resulting formulas: Next, enter the following labels and formulas for the values necessary for the sign test: G1: Count of Positive Sign

2 2 Excel Manual G2: Count of Negative Sign G3: Count of Non-Zero Difference G4: Proportion of Positive H1: =COUNTIF(E:E,1) to count the number of 1 s in column E for positives H2: =COUNTIF(E:E,-1) to count the number of -1 s for negatives H3: =H1+H2 to find the total number of signed differences H4: =H1/H3 to find the proportion of signed differences that are positive The labels and formulas for this area and an illustration of the final results are shown below: Use the information calculated in this sheet to carry out the test in text. The Wilcoxon Signed-Rank Test This example also uses the data from Example 15-7 on page 646 in the online-only portion of the text on the results of a speed writing course for secretaries. Using the spreadsheet from the previous example, insert two columns after the data and enter the following labels and formulas: F1: Absolute Difference G1: Rank I1: Sum of Positive Ranks I2: Sum of Negative Ranks

3 3 Excel Manual F2: =ABS(D2) to calculate the absolute value of the difference. G2: =RANK(F2,F:F,1) to rank the absolute differences in ascending order. J1: =SUMIF(E:E,1,G:G) to sum the ranks of positive differences J2: =SUMIF(E:E,-1,G:G) to sum the ranks of negative differences You do not need to adjust the rest of the formulas. The resulting formulas and values are shown in the following illustration: In this case, we must manually account for two situations. The first is that the difference of zero should not be ranked, so this must be removed from the column (the rankings will adjust automatically). Second, there is a tie in the rankings. We need to take the average of the ranks that are taken by the entries that tied and replace the values with this average. The next spreadsheet shows the corrections after we ve deleted the contents of F9 and ranks 4 and 5 are averaged to 4.5, which is placed into cells G2 and G6. The results are as follows: Carry out the remainder of test as shown in the example in your text.

4 4 Excel Manual Wilcoxon Rank-Sum Test Excel can be used to simplify parts of this test. This test involves giving a rank to each score based on where they would fit if the list were sorted, then summing the ranks for each treatment. =RANK(value, array_of_values, [descending(0)/ascending(1)]) ranks a list of data in ascending or descending order. Note that if there is a tie, all ties occupy the same rank. The next value occupies the rank that it would have if there were no tie. For example, if there are three values tied at the 2 nd rank, the value following the rank would be 5. The values would be ranked as {1, 2, 2, 2, 5}. =SUMIF(range,criteria,[sum_range]) sums only those values in a range that fit a specific criteria. =COMBIN(Number, Number_Chosen) calculates the combination n C r where n = number and r = number_chosen. The following example will use the data from page 654, Table 15.5, which shows the number of crimes in two different cities (City A and City B) for eight randomly selected days. Place the list of data in column A with an appropriate label in A1. Place the treatment used for that data in column B with an appropriate label in B1. Enter the label Rank in C1. A2: =RANK(A2,A:A,1) and autofill down to the last data item. As in previous examples using A:A instead of a specific range, we can enter any number of data items and not have to change the formula. We do not have use the absolute reference for the range. If you would like to use a range instead of a column, the formula would be $A$2:$A$18. This can be done automatically by using the <F4> key. Enter labels in columns E and F as in the following illustration: In F2, enter =SUMIF(B:B,E2,C:C). Autofill this down to F3 to sum the ranks for each treatment value. In F4, enter =COMBIN(COUNT(A:A),COUNTIF(B:B,"A")) to find the number of combinations of ranks possible for the two treatments.

5 5 Excel Manual The final sheet is illustrated below with values: Here is the final sheet with formulas: To correct for ties, you must manually look for ties and replace these values with the average of the ranks these values encompass. You can make this process easier by first sorting the data by rank. In this example, after sorting we can see ties for the 5 th, 8 th and 12 th ranks. That means that 12 is the rank for the 12 th and 13 th positions, so replace 12 and 13 with 12.5 in the ranking list. Likewise, replace the 8 s with 8.5 and 5 s with 5.5.

6 6 Excel Manual The final chart will look like this: The observed T is 58.5 for this test. Carry out the remainder of the test as described in the text. Kruskal-Wallis Test Calculating sums of the ranks for this test is done similarly to the previous example. The difference is that this test uses three or more samples and utilizes the Chi-Square distribution. The spreadsheet will change because there will be three or more categories and therefore three or more sums of ranks in column F instead of just two.