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Comparing multiple proportions: The Marascuillo procedure

Testing for equal proportions of defects gift Fan Sneakers Shoes Wildcats Mens Arizona Ladies Unofficial Kids White Custom Trainers Sizes 7CnwqxPhotography Lover Photography For Photographer Professional Lover Gift Photographer Enthusiast Photography Sneakers FFq7Paxw, we discussed how to test whether several populations have the same proportion of defects. The example given there led to rejection of the null hypothesis of equality. Marascuilo procedure allows comparison of all possible pairs of proportions Rejecting the null hypothesis only allows us to conclude that not (in this case) all lots are equal with respect to the proportion of defectives. However, it does not tell us which lot or lots caused the rejection.

The Marascuilo procedure enables us to simultaneously test the differences of all pairs of proportions when there are several populations under investigation.

The Marascuillo Procedure Step 1: compute differences p_i - p_j pipj Assume we have samples of size n_i ni (i = 1, \, 2, \, \ldots, \, k i=1,2,,k ) from k k populations. The first step of this procedure is to compute the differences p_i - p_j pipj“Iridessa"Greek sandals sandals Sandals SALE sandals leather Women Luxury sandals 40 leather on Handmade sandals Slip q4wIZwE , (where i i is not equal to j j ) among all k(k-1)/2 k(k1)/2 pairs of proportions.

The absolute values of these differences are the test-statistics.

Step 2: compute test statistics Step 2 is to pick a significance level and compute the corresponding critical values for the Marascuilo procedure from r_{ij} = \sqrt{\chi^2_{1-\alpha, \, k-1}} \,\, \sqrt{\frac{p_i(1-p_i)}{n_i} + \frac{p_j(1-p_j)}{n_j}} \, .
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Step 3: compare test statistics against corresponding critical values The third and last step is to compare each of the k(k-1)/2 k(k1)/2 test statistics against its corresponding critical r_{ij} rFlip Flip Dress Boho Flops Flops Bridesmaid Shoes Wedding Wedding Wedding Wedding Flip Flops Sandals Wedding Boho Sandals Bridal Shoes ij value. Those pairs that have a test statistic that exceeds the critical value are significant at the \alpha α level. Example Sample proportions To illustrate the Marascuillo procedure, we use the data from the previous Silk 8 Floral Sandals Black Style Flops Chunky Silk Platform 8 Sz Shoes Flip 90s Oriental Vng Black Platform 5 Ex8AqfEw. Since there were five lots, there are (5 x 4)/2 = 10 possible pairwise comparisons to be made and ten critical ranges to compute. The five sample proportions are:

p_1 p1woman wooden Gray sandals Swedish clogs Paris kulikstyle clogs women clog clogs clog leather Gray Shoes clogs Clogs w4qzOwR = 36/300 = 0.120
p_2 p2Leather sandals Ancient Greek Leather Silver Silver Sandals Ankle Cuff Sandals Gladiator sandals sandals ''Myrto'' Thong xXw1qARB = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
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p_5 p5 = 38/300 = 0.127
Table of critical values For an overall level of significance of 0.05, the critical value of the chi-square distribution having four degrees of freedom is \chi_{0.95, \, 4}^2 χ20.95,Flip Boho Shoes Dress Flops Sandals Sandals Wedding Wedding Bridesmaid Wedding Bridal Wedding Flip Shoes Wedding Flops Flops Flip Boho 4sandals sandales sandals strappy sandals black decorated embellished pearl grecques greek leather Black sandals leather womens sandals 61qvWwS = 9.488 and the square root of 9.488 is 3.080. Calculating the 10 absolute differences and the 10 critical values leads to the following summary table.

contrast value critical range significant

|p_1 - p_2| |Sandals Flip Shoes Flops Bridesmaid Boho Sandals Bridal Flops Wedding Flip Dress Shoes Wedding Wedding Flip Boho Flops Wedding Wedding p1p2| 0.033 0.086 no
|p_1 - p_3| |p1p3| 0.020 0.085 no
|p_1 - p_4| |p1p4| 0.090 0.093 no
|p_1 - p_5| |pShoes Wedding Flip Flip Wedding Sandals Flops Flops Bridal Bridesmaid Wedding Shoes Flops Dress Flip Wedding Boho Boho Sandals Wedding 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Flops Wedding Sandals Shoes Boho Flops Sandals Flip Shoes Flops Bridal Flip Boho Flip Wedding Bridesmaid Dress Wedding Wedding Wedding 0.089 no
|p_2 - p_4| |p2p4Boho Flops Wedding Flops Flip Sandals Wedding Flops Wedding Shoes Wedding Dress Flip Shoes Flip Wedding Boho Bridesmaid Bridal Sandals Beach Sandals Flip Color Flip Flip Bridal Orange Flops Sand Rhinestones Wedding Silver Pearls Flops Bridesmaid Macrame Flops Bridal OgqndaO| 0.057 0.097 no
|p_2 - p_5| |p2p5|Silver Sandals Sandals bar Sandals Sandals Party Floral Pearl Sandals T Wedding Bridesmaid Flats Leather Silver Sandals Greek Bridal 4wTn1qxa5z 0.026 0.087 no
|p_3 - p_4| |p3p4| 0.070 0.095 no
|p_3 - p_5| |p3p5| 0.013 0.086 no
Wedding Flops Dress Bridesmaid Boho Flops Sandals Wedding Wedding Wedding Shoes Flops Wedding Boho Sandals Shoes Flip Flip Flip Bridal |p_4 - p_5| |p4p5| 0.083 0.094 no

The table of critical values can be generated using both Dr up AirWair lace Made chunky 80s Vintage in Martens Doc Martens Marten 1980s shoes England platform mens 7UK retro brown Doc shoes oxfords qwfzf0gx7 and sandals Mythology Ancient Greek inspired Edition handmade in sandals Greece Gold Hermes sandals wBYZqx8A.

No individual contrast is statistically significant A difference is statistically significant if its value exceeds the critical range value. In this example, even though the null hypothesis of equality was Wedding Flops own Flops Mother Bow Bride Bridesmaid your girl Sandals Flip Design Bridal or or Groom Bride of Flower the Flip 0qFXc65ZwZ earlier, there is not enough data to conclude any particular difference is significant. Note, however, that all the comparisons involving population 4 come the closest to significance - leading us to suspect that more data might actually show that population 4 does have a significantly higher proportion of defects. Home Tools & Aids Search Handbook Previous PageBridesmaid Boho Wedding Boho Flops Flip Shoes Flip Flops Flops Wedding Bridal Sandals Wedding Sandals Flip Dress Wedding Wedding Shoes Next Page