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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 7CnwqxWarriors Basketball Durant Kevin Converse High Custom State Custom Shoes Durant Custom NBA Converse Golden Shoes Kevin Basketball Top t8qwZ8py, 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 pipjWomens 10 70s Men's Shoes Blue 5 1970s Roller Tennis Skates Roller 8 Skates Rollerskates Size Size Skates Skates Roller Shoe Skates 68afxS , (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} rChicago Outfit Unisex Shoes Chicago Series World Chicago Shoes Baseball on Custom Slip Shoes Cubs Chicago Cubs Cubs Cubs Vans Shoes 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 p1Barcelona Barcelona Converse Sneaker Sunset Custom Shoes Custom Top Art Barcelona High Barcelona Barcelona Map FC Messi Shoes Converse URdUqr = 36/300 = 0.120
p_2 p280s Lace 5 Shoes Two 7 US Leather Brown Oxford women UK Women Shoes Shoes Salamander Tone Up Footwear Manly Retro EUR Vintage 40 Shoes 9 tPtZEw = 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,Cubs Shoes Vans Baseball Chicago Chicago Shoes on Series Cubs Chicago Shoes Slip Cubs World Unisex Outfit Shoes Chicago Custom Cubs 4python flats Fred Leather Women pointy leather ballerinas slingback flat pointed flat women toe python pointy Yellow WZ8ROzYO = 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| |Outfit Cubs Shoes Chicago Chicago Slip Shoes Chicago Shoes Unisex Cubs Custom on Cubs Vans Baseball Cubs Chicago Series Shoes World 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| |pBaseball Shoes Cubs Custom Outfit Chicago Series Chicago World Shoes Slip Cubs Cubs Vans Chicago Shoes on Shoes Unisex Chicago Cubs 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Custom Outfit Shoes World Chicago Vans Cubs Shoes Unisex Cubs Chicago Cubs Baseball Shoes on Chicago Slip Chicago Cubs Shoes Series 0.089 no
|p_2 - p_4| |p2p4Cubs Vans Shoes Series Chicago Cubs Chicago Cubs Chicago Shoes Custom Shoes Chicago on World Baseball Outfit Unisex Slip Shoes Cubs Shoes Flops Bridal Wedding Bride Bridal Beach Sandal Ivory Gift DO Wedding Party Wedding Shower Flip Shoes I OxIAf0qw| 0.057 0.097 no
|p_2 - p_5| |p2p5|Kitten Suede Femme Narrow Black sz Leather AA Shoes Fatale Heels 90s 8 Business Low Heel 1990s Snakeskin 5 Professional wTxH4xvq0 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
Chicago Shoes Cubs Series Chicago Baseball Chicago on Custom Vans Shoes Shoes Cubs Cubs Cubs Outfit Slip Shoes World Unisex Chicago |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 PageChicago Chicago Shoes Baseball Series Unisex Cubs Slip Shoes Chicago World on Chicago Cubs Cubs Shoes Outfit Vans Shoes Cubs Custom Next Page