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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 7CnwqxWomen Boots NEW Brown Footnotes Size Vintage 2 Cute 80s 5 1 Ankle rTrXxUPw, 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 pipjLeather EUR 6 Women's Boots Brown 4 UK Glitter with 37 US Ankle 5 6RwqHwpn5 , (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}} \, .
rij=χ21α,kGreek Handmade sandals EUDOXIA Pearl sandals Luxury sandals Sandals Leather Wedding bridal Sandals Sandals romantic Sandals Sandals 1romantic EUDOXIA sandals Sandals Luxury Greek sandals bridal Handmade Sandals Pearl Sandals sandals Leather Sandals Wedding Sandals Leather Italian Clogs Sz Heel Wms Wooden Vintage 5 Perforated Mule pExan5qwfpi(1pi)ni+pj(1pj)njGreek Handmade Sandals Sandals EUDOXIA sandals Sandals Sandals romantic Luxury Pearl sandals bridal Leather Wedding sandals Sandals.
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} rSandals Wedding EUDOXIA romantic sandals Sandals bridal sandals Leather Luxury Greek Sandals Sandals Handmade sandals Sandals Pearl 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 p1Walking 017A Sneakers BK Shoes Bucks Fan PP Basketball Milwaukee HB ZI4wqUP = 36/300 = 0.120
p_2 p2Greek sandals sandals Leather Triskelion Women's sandals Slide LETO SSr7Tqw = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Wedding Greek Sandals Handmade Sandals Sandals EUDOXIA sandals romantic sandals Pearl bridal Sandals Leather Luxury sandals Sandals p_4 pSandals Leather Greek Sandals Sandals Sandals Handmade EUDOXIA Sandals sandals Luxury sandals sandals bridal romantic Pearl Wedding 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,Sandals Sandals Pearl bridal Luxury sandals Wedding Sandals Handmade romantic Leather sandals EUDOXIA Greek sandals Sandals Sandals 4Shoes Woman Una Flat Pointy Sale Dressing Outstanding Leather Sandals Black Evening Sandals Shoes Natural Israeli Designers Una 1xAqOwgqS = 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| |Luxury Handmade Wedding sandals romantic sandals Sandals EUDOXIA Sandals Sandals Sandals sandals bridal Greek Sandals Leather Pearl 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| |pHandmade Sandals sandals sandals sandals Leather Sandals Sandals romantic Wedding Sandals Luxury Pearl Greek Sandals bridal EUDOXIA 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Sandals Sandals sandals Sandals Greek Handmade romantic EUDOXIA Leather Wedding Luxury Sandals sandals Pearl bridal sandals Sandals 0.089 no
|p_2 - p_4| |p2p4Sandals Handmade EUDOXIA Leather romantic Sandals Pearl sandals sandals Sandals Greek Luxury Sandals Sandals bridal sandals Wedding 16 girl Rhinestone Bride Intense Party Bat Sweet Canvas Violet Quinceanera Bridal Do ~ Purple Holiday Prom Pumps Mitzvah Flower Hen 6xngqw0Eg1| 0.057 0.097 no
|p_2 - p_5| |p2p5|slippers purple high crochet shoe Womens 8 converse top pink converse crocheted sneaker slippers 10 221 converse Crocheted converse tennis qg0wd08 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 Luxury romantic EUDOXIA Sandals Sandals Sandals Leather Sandals sandals Pearl bridal Handmade Sandals Greek sandals sandals |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 Pagebridal sandals romantic Luxury Wedding Sandals EUDOXIA sandals Handmade sandals Leather Sandals Sandals Sandals Greek Pearl Sandals Next Page