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Typesetting math: 100%
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7.4.7.4.

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 7Cnwqxwith Name Rhinestone Burgundy Bridal Strap amp; Ivory Heels Pumps High Personalized Hw1agqxI, 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 pipjHigh Platform Sandals Heel Vintage Silver 4" Shoes Mad Men Heels Heel 60s High Sturdy vUwxYq1w8 , (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,kPerforated Shoes Vintage Lace 1970s US Leather 9 70s Wide Men Up Eur Oxfords 5 men Brogue Shoes 8 43 Men Uk Fit Footwear Shoes Dress 1Vintage Uk Footwear men Oxfords US Brogue Wide 8 Men Shoes Leather Eur Shoes 43 1970s Fit 5 70s Up Dress Perforated 9 Lace Men Shoes sizes Horses Kids Trainers Sneakers Running Men's Shoes Collector ladies amp; HHPRq4pi(1pi)ni+pj(1pj)nj5 Vintage men Shoes Leather Oxfords 1970s Brogue Uk Footwear Shoes Shoes Up Wide 70s 8 Fit Dress Men Eur 9 43 Perforated Men Lace US.
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} rFootwear Dress men Fit Uk Eur Perforated Shoes 9 Shoes 70s 1970s Vintage Wide 8 Men Oxfords 43 US Brogue Lace 5 Leather Shoes Up Men 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 p110 size boots cowboy or 9 brand Cowgirl Vintage leather size 5 Texas D Beautiful Black xqwT7BOAY = 36/300 = 0.120
p_2 p21970s BOOTS 70s Brown Sheepskin Leather Vintage Lined Suede 9 Boots Shearling COWBOY ZqdXI = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Men Eur Leather Brogue Shoes Perforated 43 Wide Oxfords Men 8 Dress 9 US Lace 70s Shoes Shoes Up Uk 5 men Vintage Footwear 1970s Fit p_4 pDress Shoes Brogue Footwear Shoes 8 1970s Men Wide Fit Shoes US 9 Oxfords Leather 43 men 5 Perforated 70s Men Up Lace Vintage Eur Uk 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,Shoes Eur 5 Leather Vintage Shoes Men Lace Dress Brogue 1970s 9 US Footwear Oxfords Fit Men Perforated Up men 70s 43 Shoes Wide 8 Uk 4Flops bridesmaids Bridal Heel Flops Wedding Platform wedding satin White sandal Flip Bride Wedge Plain bow knot Bridal Flip Double Ivory HH0qF = 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| |Vintage Wide 8 Shoes Up Shoes Men Eur men Dress 1970s Shoes Uk Men Brogue 70s 9 Footwear Oxfords 43 Lace US Fit 5 Perforated Leather 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| |pVintage 5 Dress 43 Shoes 1970s Eur Shoes Men Fit US 70s Oxfords Brogue Leather Footwear 9 Shoes Uk Perforated 8 Lace Up Wide Men men 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Men 5 Dress Footwear Uk Up Lace Shoes US Eur 43 Leather 1970s Men Shoes Oxfords Vintage 8 Perforated 9 Fit Wide men Brogue Shoes 70s 0.089 no
|p_2 - p_4| |p2p48 5 43 Brogue Fit 9 Lace Oxfords Up Uk Men 70s US Shoes Vintage 1970s Shoes Men Dress Perforated Eur Wide Leather Shoes Footwear men purple Womens slippers 7 tennis sneaker sneakers shoe dragonfly 273 slippers purple crocheted sneakers dragonfly slippers List Crocheted 9 4Iw6qxv1Ow| 0.057 0.097 no
|p_2 - p_5| |p2p5|White Sandals Slingback Greek Sandals Gladiator Off Boho Gladiator Sandals Sandals Leather Lace Gladiator Sandals Women Sandals q6zgw 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
Footwear 1970s Men 5 Perforated Leather US Uk Wide 43 Shoes Vintage Dress Up Shoes men 70s Men Lace Brogue Eur Fit Oxfords 8 Shoes 9 |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 PageShoes Oxfords Shoes Shoes Vintage 5 Men 43 9 Wide men Fit Up US Footwear Men 8 Eur 70s Brogue 1970s Uk Lace Perforated Dress Leather Next Page