Data The Heart of Statistics Bret Hanlon Department of Statistics University of Wisconsin Madison Fall 2011 Data 1 13 Purpose We need to build our statistical vocabularly For example to think about data and data analysis we need to learn how to classify variables Data 2 13 Cow Example Example A study assigned 50 cows to various diets based on the proportion of an additive in the diet and examined a number of outcomes associated with characteristics of the produced milk amount of dry matter consumed and weight gain of the cow Pre treatment variables include initial weight of the cow number of lactations and age of the cow The primary purpose of the study was to examine the effect of the different diets on the outcome variables controlling for effects of other covariates Data Case Study Example 3 13 Cow variables The variables in the data set are treatment the diet one of C ONTROL L OW M EDIUM and H IGH level the proportion of the additive in the feed lactation the number of lactations pregnancies age age of the cow at the beginning of the study in months initial weight initial weight in pounds dry mean daily weight of dry matter consumed pounds milk mean daily amount of milk produced in pounds fat percentage milk fat solids percentage of solids in milk by weight final weight final weight of cow in pounds protein percentage of protein in milk by weight Data Case Study Example 4 13 Data Here is a representative sample of the data treatment level lactation control 0 3 control 0 3 control 0 2 control 0 2 control 0 2 control 0 1 low 0 1 6 low 0 1 4 low 0 1 3 low 0 1 2 low 0 1 2 low 0 1 2 medium 0 2 3 medium 0 2 3 medium 0 2 3 medium 0 2 3 medium 0 2 2 medium 0 2 2 high 0 3 5 high 0 3 3 high 0 3 3 high 0 3 3 high 0 3 3 high 0 3 3 Data age initial weight dry milk fat solids final weight protein 49 1360 15 429 45 552 3 88 8 96 1442 3 67 47 1498 18 799 66 221 3 40 8 44 1565 3 03 36 1265 17 948 63 032 3 44 8 70 1315 3 40 33 1190 18 267 68 421 3 42 8 30 1285 3 37 31 1145 17 253 59 671 3 01 9 04 1182 3 61 22 1035 13 046 44 045 2 97 8 60 1043 3 03 89 1369 14 754 57 053 4 60 8 60 1268 3 62 74 1656 17 359 69 699 2 91 8 94 1593 3 12 45 1466 16 422 71 337 3 55 8 93 1390 3 30 34 1316 17 149 68 276 3 08 8 84 1315 3 40 36 1164 16 217 74 573 3 45 8 66 1168 3 31 41 1272 17 986 66 672 3 43 9 19 1188 3 59 45 1362 19 998 76 604 4 29 8 44 1273 3 41 49 1305 19 713 64 536 3 94 8 82 1305 3 21 48 1268 16 813 71 771 2 89 8 41 1248 3 06 44 1315 15 127 59 323 3 13 8 72 1270 3 26 40 1180 19 549 62 484 3 36 8 51 1285 3 21 35 1190 19 142 70 178 3 92 8 94 1168 3 28 81 1458 20 458 71 558 3 69 8 48 1432 3 17 49 1515 19 861 56 226 4 96 9 17 1413 3 72 48 1310 18 379 49 543 3 78 8 41 1390 3 67 46 1215 18 000 55 351 4 22 8 94 1212 3 80 49 1346 19 636 64 509 4 16 8 74 1318 3 31 46 1428 19 586 74 430 3 92 8 75 1333 3 37 Case Study Example 5 13 Categorization of variables Variables are usually either numerical quantitative or categorical qualitative numerical variables take on numerical values and are either discrete or continuous categorical variables partition the observations into categories if the categories have a natural order the variable is ordinal if not it is nominal Variables are experimental or observational experimental variables have values that are under control of the researcher observational variables have values that are observed and are not set by the researcher Variables may be response variables or explanatory variables response variables are considered as outcomes explanatory variables are thought potentially to affect outcomes Data Case Study Variables 6 13 Board Work Classify each of the variables in the cow example Data Case Study Variables 7 13 Level of Measurement Data is often represented in a rectangular array where each column is a variable and each row is an observational unit or sampling unit In the cow example individual cows are sampling units and each variable measures something on the level of the cow In other examples there may be multiple levels of measurement It is important to recognize different levels of measurement because it can affect the selection of an approriate method of analysis Data Case Study Sampling Units 8 13 Plantation Data Example Researchers interested in forest restoration in Costa Rica conducted an experiment to examine which of several species of tree best promoted the growth of native woody plants in their understory in plantations that were being converted back to natural forest The approach was to plant a fast growing native tree in the plantation that would provide shade and a suitable environment for additional native species to become established At some point the planted overstory trees would be harvested leaving a diverse natural forest behind Data Application Plantation Data 9 13 Plantation Data continued Example The study included three sites each a plantation that had been previously cleared for agriculture One site La Selva was a former experimental research station while the other two sites Paniagua and Quesada had been private farms The La Selva Paniagua and Quesada plantations were 100 m 1 3 km and 2 5 km from continuous forest respectively Each site was divided into six plots of various sizes and each plot was planted with one of six species of tree spaced in a regular array of varying sizes With minimal management the sites were allowed to grow for nearly a decade Each plot included four subplots 4m by 4m for which several variables were measured The primary response variable is the number of woody stemmed plants in each subplot Other variables include the percentage of the subplot shaded by the canopy of the overstory whether or not the subplot was flat or sloped and whether or not the subplot had good drainage Data Application Plantation Data 10 13 Sample Data stems 250 60 46 36 125 110 45 50 10 0 26 30 22 15 4 25 22 30 33 30 40 4 0 2 canopy 14 15 13 15 14 13 12 13 15 11 14 15 10 10 8 8 11 11 12 11 23 …
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