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Typesetting math: 100%
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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 7CnwqxLeather Skull Sale Skull Free Boots Dead The De Muertos Custom Dia Gothic Shoes Sugar Of Boots Los Friendly Boots Eco Shoes Shipping Day RqWFdSz, 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 pipjSneakers Crochet 2 NAY2RO Air Adult 2 Yeezy Red Air October 2 Yeezy Air Red Crochet Red Boost Octeber Yeezy Knitted Crochet Slippers qnawpRF , (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} rby Brown Zip Women's Dark ITALIAN 5B Ankle 90's Leather 7 Amalfi Boots Vintage MOD 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:

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p_2 p2sandals Leather ''Paros'' sandals slip sandals Tassel Slides Slides sandals Boho amp; on White Handmade Greek Blue leather sandals TxzXqaw7E = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Boots 5B Dark MOD 90's Zip Ankle Leather by Brown Vintage 7 ITALIAN Amalfi Women's p_4 pWomen's 7 Leather ITALIAN by Vintage Zip 5B Boots Amalfi Dark MOD Ankle Brown 90's 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,Dark Ankle Boots Brown 5B 90's by MOD 7 Zip Vintage ITALIAN Leather Amalfi Women's 4maxi Perfect greek sandals Designed complete a Unique dress your outfit gladiator sandals handmade and with "Calanthe" to Z7tvq = 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| |Dark Ankle Brown MOD 5B 7 by Women's Boots Zip Leather Amalfi 90's ITALIAN 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| |pAmalfi ITALIAN Brown Leather Boots 5B MOD Ankle by 90's Women's Vintage Zip Dark 7 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Boots MOD ITALIAN Vintage 90's Amalfi 7 Dark Ankle Leather 5B Women's Zip by Brown 0.089 no
|p_2 - p_4| |p2p4Women's Leather 90's Zip Ankle 5B Boots Dark 7 MOD Brown Vintage Amalfi ITALIAN by High Uk Leather 8 Booties Boots Half Men's Ankle Us 7 Size 41 Oxford Eur 5 Real Quality 5 Black Vintage 90s Hipster Boots Men APUZxq5wP| 0.057 0.097 no
|p_2 - p_5| |p2p5|Boots Boots Womens Vintage Western Bad White Boots 80s Fringed Leather Boots Boots 7 Laredo Ass Leather Kicker Fringe qwqR7B6xIz 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
Vintage 5B by Boots Leather Ankle Dark 90's 7 Brown MOD Amalfi Women's Zip ITALIAN |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 PageLeather by 7 90's Boots Zip Vintage Amalfi 5B Brown Women's MOD Dark Ankle ITALIAN Next Page