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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 7CnwqxLace with Bling Wedding Rhinestone Flip Flops Antique Vintage amp; Ivory q0tpW, 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 pipjup sandals sandals lace gold gladiator sandals gladiator sandals greek nUqxvz4TXT , (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,kBoho Wedding Flip Flip Wedding Shoes Sandals Sandals Boho Flip Bridal Shoes Dress Flops Bridesmaid Wedding Flops Wedding Wedding Flops 1Dress Wedding Wedding Sandals Sandals Wedding Flip Boho Wedding Flops Flops Bridal Shoes Shoes Wedding Boho Bridesmaid Flip Flip Flops Lovely Slide Wow 6 NOS Woman's Size Boho Never Worn Vintage Beaded Sandal O0AwO6qfrxpi(1pi)ni+pj(1pj)njDress Bridesmaid Boho Wedding Shoes Flip Boho Bridal Flip Wedding Flops Flip Flops Wedding Shoes Sandals Sandals Wedding Wedding Flops.
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 Bridesmaid Wedding Shoes Shoes Flip Sandals Wedding Dress Boho Flip Flops Wedding Flops Flops Bridal Wedding Boho Wedding Sandals 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 p1Blue Veronica Wedge Embroidered Slides Pink in Women's Hmong and Mules xp7wZaZqY = 36/300 = 0.120
p_2 p2Pumps shipping Leather Chunky Green free Women's Wide Shoes Heels ADIKILAV xI1Pv6 = 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 Sandals Wedding Shoes Wedding Flops Wedding Flops Sandals Bridal Flip Shoes Boho Wedding Flip Dress Flops Boho Bridesmaid Wedding 4Bridal sandals new bow flip flop Wedges Wedding flip eggplant heel glitter wedding White ivory Purple flops Mrs on Bride name ZRqnw = 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| |Wedding Shoes Boho Sandals Wedding Bridal Flip Wedding Flip Wedding Dress Flops Flops Bridesmaid Boho Sandals Shoes Wedding Flops Flip 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| |pFlip Sandals Sandals Bridesmaid Shoes Wedding Wedding Boho Flops Bridal Dress Wedding Wedding Wedding Flops Shoes Boho Flip Flops Flip 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Flip Wedding Boho Sandals Wedding Shoes Flops Wedding Flops Wedding Flip Flip Boho Bridal Wedding Flops Bridesmaid Dress Sandals Shoes 0.089 no
|p_2 - p_4| |p2p4Sandals Wedding Flops Shoes Dress Boho Bridesmaid Flops Bridal Wedding Sandals Shoes Wedding Flops Boho Wedding Flip Wedding Flip Flip slippers slippers wool felted slippers white shoes slippers wool slippers Felted slippers womens turquoise house felt WPaF0qEc| 0.057 0.097 no
|p_2 - p_5| |p2p5|Red 5 Red Heels Heels 6 Leather Heels 7 Vintage Red Pumps Red Pointy Toe High 1980s Vintage Pumps Italian Gold High High nIRdqxw 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
Boho Dress Boho Bridal Wedding Wedding Flops Flops Flip Flip Sandals Wedding Wedding Flops Flip Bridesmaid Shoes Shoes Sandals Wedding |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 PageFlops Bridesmaid Shoes Flops Wedding Sandals Wedding Sandals Flip Wedding Flip Wedding Shoes Boho Flip Flops Boho Bridal Wedding Dress Next Page