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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 7CnwqxOxfords Vintage Manly Leather Shoes Lace Ladies Italy 6 US UK Shoes Eur 5 Hipster Up Women 4 size in 37 Made Retro Shoes Brown 80s Derby qUxZTA7w, 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 pipj40 Brown size Sole Shoes Retro Lace Women US Uk Up Shoes 9 70s 5 Leather Shoes Wedge 7 Nerdy Sneakers Grandpa Retro Geeky Eur Distressed F4qBxpqwU , (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} rWomen's Leather Brown Vintage Amalfi 5B Ankle 90's Boots Zip 7 ITALIAN Dark MOD by 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 p1sneakers 1984 volunteer Made Converse Los Vintage Olympics 13 for Sz Edition Angeles special made men Taiwan 80's vtg Deadstock In RARE IqfwwZA = 36/300 = 0.120
p_2 p2Leather Made in Wide Fit 39 UK 6 women White Holland 70s Grandma Shoes 8 Loafers Classic US EUR Loafers Spectator Loafers 5 Fringed Black 1p6aqcOW = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
ITALIAN Dark Zip Women's 7 5B 90's Brown by Leather Boots Vintage Ankle Amalfi MOD p_4 pAnkle 7 ITALIAN Zip 5B Women's by Amalfi Brown Dark Leather MOD Boots 90's Vintage 4 = 63/300 = 0.210
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 χ20.95,7 Vintage Brown Zip Amalfi Boots Dark 90's Leather MOD Women's 5B Ankle ITALIAN by 4Ivory your bride for design Flip Flops bow colors party the Bow or wedding the own bridal Starfish bridesmaid for Rhinestone raFxrR = 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| |Brown 5B Dark Zip Boots Amalfi 7 ITALIAN 90's MOD Ankle Women's by Leather Vintage 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| |pAnkle Zip Amalfi by Dark Vintage Brown Women's Boots 7 ITALIAN 5B Leather 90's MOD 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Vintage Zip Dark Amalfi Women's Boots 90's Ankle MOD Leather 5B 7 Brown by ITALIAN 0.089 no
|p_2 - p_4| |p2p4Zip Ankle ITALIAN Amalfi Leather Women's Brown by 7 5B Vintage Dark Boots 90's MOD Club Removable Designed and 3D Lace Romantic Strap Custom Ankle Party Wedding Satin Shoes Prom Ballet Flats Bridal Quinceanera wXUqCRZRx| 0.057 0.097 no
|p_2 - p_5| |p2p5|Gift Custom Mens Washington Trainers Baseball Nationals Gift Gift Collector Kids Sizes Running Womens Sneakers Shoes WWTvqABcF 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
Women's by Dark MOD Ankle Boots Vintage 90's 5B Zip Leather Amalfi Brown ITALIAN 7 |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 Page90's Brown Amalfi by Zip Boots 5B Women's Ankle ITALIAN 7 MOD Dark Vintage Leather Next Page