Testing Assumptions and Exploring Data
Conducting data analysis is like drinking a fine wine. It is important to swirl and sniff the wine, to unpack the complex bouquet and to appreciate the experience. Gulping the wine doesn’t work. [Wright, D.B. 2003. Making friends with your data: Improving how statistics are conducted and reported. British Journal of Educational Psychology, 73, 123-136.]
Most statistical tests and techniques have underlying assumptions that are often violated (i.e., the data or model do not meet the assumptions). Some of these violations have little impact on the results or conclusions.
Others can increase type I (false positive) or type II (false negative) errors, potentially leading to wrong conclusions and erroneous recommendations. Most statistical violations can be avoided by exploring your data better before analysis.
In this lesson, we will examine graphical and statistical methods for data exploration and testing assumptions. We follow the recommendations of Zuur et al., 2010. A protocol for data exploration to avoid common statistical problems. Methods in Ecology and Evolution 1, 3–14.
All models and statistical tests have assumptions. Different tests can deal with violation of some assumptions better than others. For example, heterogeneity (differences in variation) may be problematic in linear regression, or a single outlier may exert a huge influence on estimates of the mean.
These assumptions come under three broad categories: (1) Distributional assumptions are concerned with the probability distributions of the observations or their associated random errors (e.g., normal versus binomial), (2) Structural assumptions are concerned with the form of the functional relationships between the response variable and the predictor variables (e.g., linear regression assumes that the relationship is linear), (3) Cross-variation assumptions are concerned with the joint probability distribution of the observations and/or the errors (e.g., most tests assume that the observations are independent).
Before we proceed, two other important issue to be clear about.
First, data exploration is a very different, and separate, process from hypothesis testing (or Bayesian, likelihood, or information theoretic approaches). Models and hypotheses should be based on your understanding of your study system, not on ‘data dredging’ looking for patterns.
Second, here we emphasize the visual inspection of the data and model, rather than an over-reliance on statistical tests of the assumptions. There are statistical tests for normality and homogeneity, but the statistical literature warns against them. See the Best Practice and Resources pages for more information.
We will explore several data sets now. First, look at the head of the sparrow dataset loaded with this lesson. The dataset is called ‘sparrows’
head(sparrows)## Species Sex Wing Tarsus Head Culmen Nalospi Weight Observer Age
## 1 SSTS Male 58.0 21.7 32.7 13.9 10.2 20.3 2 0
## 2 SSTS Female 56.5 21.1 31.4 12.2 10.1 17.4 2 0
## 3 SSTS Male 59.0 21.0 33.3 13.8 10.0 21.0 2 0
## 4 SSTS Male 59.0 21.3 32.5 13.2 9.9 21.0 2 0
## 5 SSTS Male 57.0 21.0 32.5 13.8 9.9 19.8 2 0
## 6 SSTS Female 57.0 20.7 32.5 13.3 9.9 17.5 2 0You can see 10 columns with morphometric information on 979 individual sparrows from females and males of two species.
Are there outliers in Y and X?
An outlier is an observation or data point that is ‘abnormally’ high or low compared to the other data. It may indicate real variation or experimental, observation, and/or data entry errors. As such, the definition of what constitutes an outlier is subjective.
Different techniques respond to and treat outliers differently. For some analyses, outliers make no difference; for others they may bias the results and conclusions.
The classic graph for looking at outliers is the boxplot, which we have seen before. Make a simple boxplot of sparrow wing length, ‘sparrows$Wing’.
boxplot(sparrows$Wing)
The bold line in the middle of the box indicates the median, the lower end of the box is the 25% quartile, the upper end of the box is the 75% quartile. The ‘hinge’ is the 75% quartile minus the 25% quartile. The lines (or whiskers) indicate 1.5 times the size of the hinge. (Note that the interval defined by these lines is not a confidence interval.) Points beyond these lines are (often wrongly) considered to be outliers.
In this case, the boxplot is suggesting to us that there may be at least 6 outliers.
Boxplots provide a summary of the data. As we will discuss in Unit 5, a better approach would be to present the raw data as well. We can use the basic plot() function to do this. Use plot() to display the wing measurements.
plot(sparrows$Wing)
As usual, running plot() on a numeric vector creates a plot showing each data point as an open circle, displayed in the order they occur in the vector (indicated by the x-axis, labelled Index).
This plot is a version of the ‘Cleveland dotplot’, the row number (i.e., index) of an observation is plotted vs. the observation value.
R has the function dotchart() to do this better than our simple use ofplot(). Rundotchart() on the same sparrow wing data.
dotchart(sparrows$Wing)
Here the function has rotated the plot, compared to before. The data are organised by index on the y-axis and the values are indicated on the x-axis. We also see vertical grey lines that make it easier to compare the horizontal position of each point.
This dotplot suggests that there may in fact be fewer than 6 outliers. A nice feature of this function is that we can condition the continuous variable on other factors in the data, using the argument ‘group =’.
Create a dotchart of the sparrow wing data, grouped by species.
dotchart(sparrows$Wing, group = sparrows$Species)
Now we can see that most of the ‘outliers’ in fact belong to the species ‘SESP’ and appear much more within the expected range of the data. There is only one value from the ‘SSTS’ species that appears abnormally high.
What you then do with any outliers depends. If there are no data entry errors, you could check for other explanatory factors, transform the data, or drop the sample with the outlier, depending on the sensitivity of the analysis. Be careful how you treat outliers - ethically removing points of data because they don’t fit your expectation requires strong justification.
Do we have homogeneity of variance?
Homogeneity of variance is an important assumption in analysis of variance (ANOVA), other regression-related models and some multivariate techniques. Violation of constant variance across groups makes it hard to estimate the standard deviation of the coefficient estimates.
In ANOVA, we can check the variance by making conditional boxplots for each group. Make a boxplot of $Wing conditional on $Species and \(Sex. We can do this by writing a formula: sparrows\)Wing ~ sparrows\(Sex + sparrows\)Species. The tilde (~) means ‘as a function of’. Create a plot of boxplots using this formula.
boxplot(sparrows$Wing ~ sparrows$Sex + sparrows$Species)
To perform this model as an ANOVA, the variation in the observations from the sexes should be similar, as should the variation in the observations from the species. In this case, there seems to be less variation in females of SSTS than females of SESP. Larger differences would be more worrying.
To verify that variances are homogeneous in regression-type models with continuous predictors, we should use the residuals (i.e., the differences between the observed values and the estimated values) of the model. We have loaded a model of sparrow wing length on weight + species. Type ‘m’ to look at the model coefficients.
m##
## Call:
## lm(formula = sparrows$Wing ~ sparrows$Weight + sparrows$Species)
##
## Coefficients:
## (Intercept) sparrows$Weight sparrows$SpeciesSSTS
## 43.6014 0.7476 -0.9747Now type ‘str(m)’ to look at the model structure and see how we can extract the residuals.
str(m)## List of 13
## $ coefficients : Named num [1:3] 43.601 0.748 -0.975
## ..- attr(*, "names")= chr [1:3] "(Intercept)" "sparrows$Weight" "sparrows$SpeciesSSTS"
## $ residuals : Named num [1:979] 0.197 0.865 0.673 0.673 -0.43 ...
## ..- attr(*, "names")= chr [1:979] "1" "2" "3" "4" ...
## $ effects : Named num [1:979] -1810.567 47.154 7.991 0.662 -0.448 ...
## ..- attr(*, "names")= chr [1:979] "(Intercept)" "sparrows$Weight" "sparrows$SpeciesSSTS" "" ...
## $ rank : int 3
## $ fitted.values: Named num [1:979] 57.8 55.6 58.3 58.3 57.4 ...
## ..- attr(*, "names")= chr [1:979] "1" "2" "3" "4" ...
## $ assign : int [1:3] 0 1 2
## $ qr :List of 5
## ..$ qr : num [1:979, 1:3] -31.289 0.032 0.032 0.032 0.032 ...
## .. ..- attr(*, "dimnames")=List of 2
## .. .. ..$ : chr [1:979] "1" "2" "3" "4" ...
## .. .. ..$ : chr [1:3] "(Intercept)" "sparrows$Weight" "sparrows$SpeciesSSTS"
## .. ..- attr(*, "assign")= int [1:3] 0 1 2
## .. ..- attr(*, "contrasts")=List of 1
## .. .. ..$ sparrows$Species: chr "contr.treatment"
## ..$ qraux: num [1:3] 1.03 1.05 1.02
## ..$ pivot: int [1:3] 1 2 3
## ..$ tol : num 1e-07
## ..$ rank : int 3
## ..- attr(*, "class")= chr "qr"
## $ df.residual : int 976
## $ contrasts :List of 1
## ..$ sparrows$Species: chr "contr.treatment"
## $ xlevels :List of 1
## ..$ sparrows$Species: chr [1:2] "SESP" "SSTS"
## $ call : language lm(formula = sparrows$Wing ~ sparrows$Weight + sparrows$Species)
## $ terms :Classes 'terms', 'formula' language sparrows$Wing ~ sparrows$Weight + sparrows$Species
## .. ..- attr(*, "variables")= language list(sparrows$Wing, sparrows$Weight, sparrows$Species)
## .. ..- attr(*, "factors")= int [1:3, 1:2] 0 1 0 0 0 1
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:3] "sparrows$Wing" "sparrows$Weight" "sparrows$Species"
## .. .. .. ..$ : chr [1:2] "sparrows$Weight" "sparrows$Species"
## .. ..- attr(*, "term.labels")= chr [1:2] "sparrows$Weight" "sparrows$Species"
## .. ..- attr(*, "order")= int [1:2] 1 1
## .. ..- attr(*, "intercept")= int 1
## .. ..- attr(*, "response")= int 1
## .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
## .. ..- attr(*, "predvars")= language list(sparrows$Wing, sparrows$Weight, sparrows$Species)
## .. ..- attr(*, "dataClasses")= Named chr [1:3] "numeric" "numeric" "factor"
## .. .. ..- attr(*, "names")= chr [1:3] "sparrows$Wing" "sparrows$Weight" "sparrows$Species"
## $ model :'data.frame': 979 obs. of 3 variables:
## ..$ sparrows$Wing : num [1:979] 58 56.5 59 59 57 57 57 57 53.5 56.5 ...
## ..$ sparrows$Weight : num [1:979] 20.3 17.4 21 21 19.8 17.5 19.6 21.2 18.5 20.5 ...
## ..$ sparrows$Species: Factor w/ 2 levels "SESP","SSTS": 2 2 2 2 2 2 2 2 2 2 ...
## ..- attr(*, "terms")=Classes 'terms', 'formula' language sparrows$Wing ~ sparrows$Weight + sparrows$Species
## .. .. ..- attr(*, "variables")= language list(sparrows$Wing, sparrows$Weight, sparrows$Species)
## .. .. ..- attr(*, "factors")= int [1:3, 1:2] 0 1 0 0 0 1
## .. .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. .. ..$ : chr [1:3] "sparrows$Wing" "sparrows$Weight" "sparrows$Species"
## .. .. .. .. ..$ : chr [1:2] "sparrows$Weight" "sparrows$Species"
## .. .. ..- attr(*, "term.labels")= chr [1:2] "sparrows$Weight" "sparrows$Species"
## .. .. ..- attr(*, "order")= int [1:2] 1 1
## .. .. ..- attr(*, "intercept")= int 1
## .. .. ..- attr(*, "response")= int 1
## .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
## .. .. ..- attr(*, "predvars")= language list(sparrows$Wing, sparrows$Weight, sparrows$Species)
## .. .. ..- attr(*, "dataClasses")= Named chr [1:3] "numeric" "numeric" "factor"
## .. .. .. ..- attr(*, "names")= chr [1:3] "sparrows$Wing" "sparrows$Weight" "sparrows$Species"
## - attr(*, "class")= chr "lm"There is a lot in there! But, scroll up to the top and you will see $ residuals : Named num [1:979] 0.197 .... Extract (subset) this part of m as a vector. Remember that m, the model output, is a list.
m$residuals## 1 2 3 4 5
## 0.1965088231 0.8646123991 0.6731734771 0.6731734771 -0.4296802155
## 6 7 8 9 10
## 1.2898502068 -0.2801558310 -1.4763509074 -2.9577717160 -1.4530155615
## 11 12 13 14 15
## -4.4249240611 -1.5325339082 -1.1025399460 1.4488869003 0.3226978617
## 16 17 18 19 20
## -0.8596741777 1.8693685536 1.4627099373 -0.8268265229 -0.1539667924
## 21 22 23 24 25
## 1.5236490926 0.8974600540 1.0936551304 -0.7101497932 2.2479356694
## 26 27 28 29 30
## 0.0236490926 2.2712710153 0.5469844385 -0.7753996765 0.9488869003
## 31 32 33 34 35
## 1.4955575922 1.1217466308 -0.0558692542 -0.7801558310 -2.7753996765
## 36 37 38 39 40
## -1.6025399460 -0.0558692542 -0.3268265229 -1.2006374842 -0.6539667924
## 41 42 43 44 45
## -0.7753996765 0.0141367836 -1.9435032525 2.2712710153 2.2712710153
## 46 47 48 49 50
## 1.2712710153 -1.1306314464 0.2665148609 -0.3549180233 1.4207953999
## 51 52 53 54 55
## 2.2150880145 -0.7849119855 1.4955575922 0.4207953999 0.8974600540
## 56 57 58 59 60
## -2.3501618688 2.6731734771 2.2712710153 0.8226978617 0.6731734771
## 61 62 63 64 65
## 0.4441307458 1.0469844385 0.5750759389 2.6450819767 0.7712710153
## 66 67 68 69 70
## -0.6258752920 2.8974600540 1.3741247080 0.8226978617 2.0236490926
## 71 72 73 74 75
## 1.0141367836 0.1684173227 -1.4111010241 0.5984112849 -1.1120522550
## 76 77 78 79 80
## -0.5887169091 -1.2006374842 1.1965088231 0.5984112849 0.7993625158
## 81 82 83 84 85
## 0.8974600540 0.8974600540 2.2479356694 1.7198441690 -1.8501618688
## 86 87 88 89 90
## 0.9955575922 -1.5277777538 2.1498381312 1.1217466308 -0.5511130997
## 91 92 93 94 95
## 0.7760271698 -0.8549180233 1.0703197845 1.4488869003 0.5703197845
## 96 97 98 99 100
## 0.3741247080 1.8974600540 -1.0372900627 0.0374721296 -4.1025399460
## 101 102 103 104 105
## -0.5325339082 -0.9811070619 -1.7849119855 1.4207953999 -0.4111010241
## 106 107 108 109 110
## -1.0558692542 1.0936551304 -3.6820582928 -1.1587229469 -2.2287289847
## 111 112 113 114 115
## 3.3226978617 -0.1025399460 1.1498381312 3.1965088231 2.7946063613
## 116 117 118 119 120
## -2.8034911769 -0.8830095237 1.0469844385 -1.2334851391 2.4207953999
## 121 122 123 124 125
## 2.7760271698 2.4207953999 0.2712710153 -0.9577717160 0.7665148609
## 126 127 128 129 130
## 0.2946063613 0.4441307458 0.2198441690 0.1217466308 2.3974600540
## 131 132 133 134 135
## -0.0511130997 0.2712710153 -0.7053936387 1.5236490926 -0.4296802155
## 136 137 138 139 140
## 2.5984112849 -0.6773021383 -0.1306314464 -0.1773021383 1.2246003235
## 141 142 143 144 145
## -0.6820582928 0.8974600540 -1.0839607546 2.8460332076 2.4393745914
## 146 147 148 149 150
## 0.0655636300 -2.1587229469 -1.1587229469 -0.0511130997 2.2712710153
## 151 152 153 154 155
## 2.3460332076 1.9674660918 0.2712710153 2.3693685536 -0.0744484456
## 156 157 158 159 160
## -2.9996862533 1.4207953999 1.9627099373 0.3226978617 2.7946063613
## 161 162 163 164 165
## 0.0750759389 0.8226978617 0.1498381312 2.4207953999 2.2993625158
## 166 167 168 169 170
## 1.4207953999 1.4955575922 -3.7287289847 -1.7849119855 2.6450819767
## 171 172 173 174 175
## 1.7198441690 1.1965088231 -0.0839607546 0.9488869003 -1.9063448696
## 176 177 178 179 180
## -0.7287289847 -0.6072961005 3.0188929381 -2.2334851391 -2.5325339082
## 181 182 183 184 185
## -0.6820582928 2.5422282840 -4.4715947529 -1.8830095237 1.5422282840
## 186 187 188 189 190
## -1.4763509074 1.4207953999 -3.3268265229 0.0936551304 0.2150880145
## 191 192 193 194 195
## 1.8974600540 1.3693685536 1.5236490926 2.5984112849 3.3460332076
## 196 197 198 199 200
## -1.4296802155 2.2993625158 0.0003137467 0.2993625158 3.4722222462
## 201 202 203 204 205
## 0.9207953999 2.1684173227 3.1684173227 -0.8082473314 1.9112830909
## 206 207 208 209 210
## -0.1120522550 -2.8315826773 -3.3734972147 -1.9811070619 -0.9811070619
## 211 212 213 214 215
## -2.5277777538 0.3179417072 -0.5325339082 -1.1492106379 0.7479356694
## 216 217 218 219 220
## 0.7993625158 -0.8315826773 2.9441307458 0.8226978617 0.5236490926
## 221 222 223 224 225
## 1.3974600540 3.1731734771 0.0003137467 -0.5511130997 1.3460332076
## 226 227 228 229 230
## -1.6587229469 -1.0091985623 -0.7287289847 0.2898502068 0.9488869003
## 231 232 233 234 235
## -1.3830095237 0.5655636300 0.1965088231 0.1217466308 -2.7287289847
## 236 237 238 239 240
## 1.8974600540 0.0188929381 0.7760271698 1.8693685536 -3.5792046001
## 241 242 243 244 245
## 0.6122343218 2.9722222462 -2.4763509074 1.6450819767 0.5422282840
## 246 247 248 249 250
## -1.1773021383 1.1122343218 0.0188929381 -1.1306314464 -0.9811070619
## 251 252 253 254 255
## 1.6965088231 0.1965088231 -0.1306314464 -0.4811070619 0.0469844385
## 256 257 258 259 260
## -3.5744484456 -1.7101497932 0.5703197845 1.1684173227 -3.1773021383
## 261 262 263 264 265
## 2.9908014377 -1.2101497932 -1.1472147654 3.1565901582 -0.6472147654
## 266 267 268 269 270
## 2.0023096192 -1.3248306504 0.5584926200 1.2827791968 0.3294498887
## 271 272 273 274 275
## 4.7080170045 1.6237425033 -0.2267331122 0.6704131952 0.5584926200
## 276 277 278 279 280
## -2.0957879191 1.4742181187 1.0770718114 0.6799255041 3.1799255041
## 281 282 283 284 285
## 1.6799255041 -3.7219769577 0.0537364655 -3.2734038040 2.1284986578
## 286 287 288 289 290
## -2.6753062658 -2.4181720341 5.7265961960 3.5770718114 0.9789742732
## 291 292 293 294 295
## -1.7686476495 -0.0491172272 0.6518340037 -3.1753062658 1.4508827728
## 296 297 298 299 300
## -1.2453123036 -0.3948366882 1.2479356694 -1.2896681400 -0.4249240611
## 301 302 303 304 305
## -0.1773021383 1.0003137467 2.1217466308 0.3226978617 -0.8501618688
## 306 307 308 309 310
## 0.3741247080 0.9160392454 -1.7006374842 1.1217466308 0.6403258223
## 311 312 313 314 315
## 1.6731734771 -1.3782533692 2.4207953999 -1.4296802155 -0.1773021383
## 316 317 318 319 320
## -0.9811070619 -2.3830095237 -2.0792046001 -0.0792046001 -0.5839607546
## 321 322 323 324 325
## 0.5703197845 0.0469844385 0.5984112849 1.2479356694 -1.0558692542
## 326 327 328 329 330
## 1.2479356694 -0.2053936387 0.0188929381 0.3460332076 0.3741247080
## 331 332 333 334 335
## 1.9207953999 -0.7334851391 1.1731734771 -1.3268265229 -1.5325339082
## 336 337 338 339 340
## -2.1306314464 0.8693685536 0.0188929381 1.8226978617 -0.0277777538
## 341 342 343 344 345
## -0.6539667924 -0.2053936387 0.8226978617 1.0469844385 -1.2239728302
## 346 347 348 349 350
## 1.0236490926 -0.4811070619 0.7198441690 -1.3082473314 1.6684173227
## 351 352 353 354 355
## 1.1731734771 1.1217466308 1.8974600540 1.1965088231 -5.0744484456
## 356 357 358 359 360
## -1.2053936387 -1.4015887151 2.2617587064 0.1450819767 -3.1773021383
## 361 362 363 364 365
## -1.9763509074 0.2712710153 -0.8782533692 2.5469844385 1.8460332076
## 366 367 368 369 370
## -0.9344363700 0.5236490926 2.4955575922 2.5236490926 1.4207953999
## 371 372 373 374 375
## 0.9255515544 -0.9530155615 -0.2053936387 0.9722222462 -1.7101497932
## 376 377 378 379 380
## -0.7287289847 0.5236490926 1.3460332076 -2.0792046001 -0.2053936387
## 381 382 383 384 385
## -3.0277777538 1.1217466308 -1.5230215993 -0.2801558310 -1.9296802155
## 386 387 388 389 390
## 0.4955575922 0.4955575922 -0.0277777538 -0.1773021383 0.8131855527
## 391 392 393 394 395
## 2.5703197845 2.0188929381 0.6450819767 0.3507893621 -1.4811070619
## 396 397 398 399 400
## 0.9955575922 -1.8782533692 -2.5511130997 1.1217466308 -1.3315826773
## 401 402 403 404 405
## -3.9249240611 1.2946063613 -0.2006374842 1.1965088231 -1.4577717160
## 406 407 408 409 410
## 1.5236490926 1.0984112849 0.0469844385 1.1450819767 -2.1773021383
## 411 412 413 414 415
## 0.5141367836 -0.3549180233 1.4955575922 0.6450819767 -3.1025399460
## 416 417 418 419 420
## -1.3363388318 -2.2801558310 -2.4858632164 -2.0558692542 0.3460332076
## 421 422 423 424 425
## 0.7993625158 -2.6539667924 -2.4577717160 -2.4063448696 -1.4577717160
## 426 427 428 429 430
## 1.0188929381 0.1217466308 0.9627099373 0.1450819767 -3.4763509074
## 431 432 433 434 435
## -0.1025399460 1.0936551304 1.6965088231 -1.9530155615 1.0469844385
## 436 437 438 439 440
## 0.3741247080 1.7946063613 -0.4625278704 -2.4811070619 -1.1025399460
## 441 442 443 444 445
## 2.3179417072 -0.8315826773 -0.4625278704 -3.7334851391 0.5469844385
## 446 447 448 449 450
## 0.0936551304 -1.5792046001 -0.6353876009 -2.4763509074 -1.7101497932
## 451 452 453 454 455
## -0.1820582928 0.5703197845 -2.4296802155 2.6450819767 2.6684173227
## 456 457 458 459 460
## -1.1025399460 1.8460332076 0.1965088231 0.1965088231 -2.6820582928
## 461 462 463 464 465
## 1.7712710153 -0.7006374842 0.0936551304 -1.3549180233 0.2946063613
## 466 467 468 469 470
## 0.5703197845 -0.9625278704 2.9441307458 1.7946063613 0.8226978617
## 471 472 473 474 475
## 0.0236490926 1.1450819767 2.6731734771 0.9722222462 3.2384233605
## 476 477 478 479 480
## 0.0188929381 -0.6539667924 2.3741247080 2.0888989759 -2.2006374842
## 481 482 483 484 485
## 0.3646123991 -0.1725459838 0.3460332076 -0.7287289847 0.9955575922
## 486 487 488 489 490
## -0.9530155615 0.0469844385 0.0236490926 0.2150880145 -0.4858632164
## 491 492 493 494 495
## -2.6820582928 1.2712710153 -0.2615766395 1.8693685536 -1.2334851391
## 496 497 498 499 500
## -2.9715947529 -1.7801558310 -0.5277777538 -0.5792046001 -1.7006374842
## 501 502 503 504 505
## -4.9530155615 1.4955575922 -1.7334851391 1.3460332076 -0.2801558310
## 506 507 508 509 510
## 0.1217466308 1.4955575922 0.2946063613 -1.9811070619 -2.2334851391
## 511 512 513 514 515
## 2.4207953999 0.1731734771 -0.1143671105 -2.5491172272 -1.5257818813
## 516 517 518 519 520
## -1.2734038040 -3.0443610727 -3.4557758435 1.5584926200 -2.3762574967
## 521 522 523 524 525
## -0.7686476495 -2.2919829955 1.0537364655 0.8013583883 -0.2967391500
## 526 527 528 529 530
## 0.7080170045 3.0770718114 0.2966022338 0.1284986578 0.3061145427
## 531 532 533 534 535
## 3.0770718114 -1.1800624203 -3.4976903808 -0.5443610727 0.4275474269
## 536 537 538 539 540
## 1.5770718114 -0.3995928426 -4.3200744959 1.0469844385 1.3460332076
## 541 542 543 544 545
## -0.0277777538 -0.0277777538 -0.9063448696 1.2712710153 0.5422282840
## 546 547 548 549 550
## -1.9811070619 -0.7287289847 2.4207953999 1.0469844385 -0.3268265229
## 551 552 553 554 555
## -3.7287289847 -0.2053936387 -1.4577717160 -0.7101497932 0.9255515544
## 556 557 558 559 560
## 0.1684173227 -2.6820582928 1.5469844385 -1.2053936387 -0.9811070619
## 561 562 563 564 565
## -1.8315826773 -0.5792046001 1.5141367836 -1.8549180233 -0.7006374842
## 566 567 568 569 570
## 2.2993625158 1.2712710153 -3.6958813298 -1.2334851391 -1.8315826773
## 571 572 573 574 575
## 0.2712710153 0.7198441690 -3.1773021383 1.9955575922 0.7946063613
## 576 577 578 579 580
## -2.5091985623 1.8693685536 1.3179417072 2.2198441690 1.7946063613
## 581 582 583 584 585
## -2.1072961005 -0.4577717160 -0.1587229469 -0.6820582928 -0.9811070619
## 586 587 588 589 590
## 2.0374721296 1.0188929381 2.0469844385 -0.1306314464 -0.9530155615
## 591 592 593 594 595
## 0.5236490926 -0.2053936387 0.2712710153 1.6684173227 0.7898502068
## 596 597 598 599 600
## 0.1731734771 -4.9811070619 3.0422282840 0.0188929381 0.3226978617
## 601 602 603 604 605
## -0.1773021383 -0.4296802155 0.2712710153 -0.0091985623 2.2712710153
## 606 607 608 609 610
## 0.6731734771 0.4255515544 0.0188929381 -2.9063448696 -0.4134158796
## 611 612 613 614 615
## -0.6657939569 2.3808767350 1.0537364655 -0.9462635345 -1.6657939569
## 616 617 618 619 620
## 3.0070657736 1.4556389273 1.8061145427 -1.7033977662 2.3246937342
## 621 622 623 624 625
## -1.6753062658 0.7946063613 -0.2239728302 1.5703197845 -1.6820582928
## 626 627 628 629 630
## -0.6539667924 1.5236490926 -0.8315826773 -1.0839607546 -0.8315826773
## 631 632 633 634 635
## -2.0839607546 -0.9530155615 -2.2053936387 -2.4577717160 -0.7287289847
## 636 637 638 639 640
## -1.3363388318 1.9722222462 -0.7101497932 -0.6820582928 0.5236490926
## 641 642 643 644 645
## 0.0469844385 1.0469844385 -0.5792046001 -0.9530155615 2.0750759389
## 646 647 648 649 650
## 1.5703197845 -0.9530155615 -0.0792046001 10.7946063613 -1.4763509074
## 651 652 653 654 655
## -2.8315826773 -1.2053936387 1.1684173227 -1.1306314464 1.7946063613
## 656 657 658 659 660
## -0.2053936387 -0.3363388318 0.0469844385 -0.4577717160 -1.2101497932
## 661 662 663 664 665
## 1.9112830909 -0.9249240611 1.6450819767 -3.7006374842 -1.9811070619
## 666 667 668 669 670
## 0.3342060431 2.1731734771 0.7479356694 -4.2753996765 0.9722222462
## 671 672 673 674 675
## 1.1355696678 -1.0839607546 0.3412770531 -0.7520643306 0.0469844385
## 676 677 678 679 680
## 0.2946063613 -7.1163629830 2.2617587064 1.7946063613 0.0374721296
## 681 682 683 684 685
## 1.4674660918 -3.9530155615 -0.1587229469 -0.8877656782 0.4207953999
## 686 687 688 689 690
## 1.4674660918 0.7946063613 1.7712710153 0.7246003235 -2.3596741777
## 691 692 693 694 695
## -3.2006374842 0.0469844385 1.7665148609 0.1450819767 -0.5792046001
## 696 697 698 699 700
## 1.9207953999 -2.4530155615 0.0469844385 -0.7801558310 2.4207953999
## 701 702 703 704 705
## -1.2334851391 1.9207953999 -0.5044424078 0.6450819767 -0.7287289847
## 706 707 708 709 710
## -1.8501618688 -1.2053936387 -0.0839607546 1.6684173227 -1.7849119855
## 711 712 713 714 715
## -2.2006374842 2.0469844385 -1.8315826773 -2.7006374842 -2.2615766395
## 716 717 718 719 720
## -1.3268265229 1.0469844385 0.9722222462 -0.9530155615 -2.0277777538
## 721 722 723 724 725
## 0.9255515544 1.5936551304 0.7665148609 0.7198441690 0.0469844385
## 726 727 728 729 730
## -1.4577717160 -0.9530155615 1.0469844385 -2.0091985623 0.4160392454
## 731 732 733 734 735
## 2.0469844385 1.2712710153 0.5517405930 1.3460332076 -1.3596741777
## 736 737 738 739 740
## -0.1539667924 -1.4530155615 0.0188929381 0.9955575922 -0.2053936387
## 741 742 743 744 745
## 0.2431795149 0.5469844385 0.7993625158 -1.2006374842 2.2712710153
## 746 747 748 749 750
## -1.0792046001 -1.5792046001 -0.9530155615 0.1684173227 -1.2053936387
## 751 752 753 754 755
## -3.8315826773 -0.7053936387 0.0469844385 -0.2053936387 -1.4858632164
## 756 757 758 759 760
## -0.4811070619 0.2946063613 -1.3596741777 -5.5744484456 0.5236490926
## 761 762 763 764 765
## -2.0744484456 1.7898502068 -2.3549180233 0.0936551304 1.3927038995
## 766 767 768 769 770
## -0.9530155615 -0.0744484456 -2.7287289847 -1.2239728302 -1.9249240611
## 771 772 773 774 775
## -1.4482594070 1.5703197845 0.6731734771 1.7993625158 -0.2801558310
## 776 777 778 779 780
## -0.3268265229 0.1731734771 2.3179417072 0.9207953999 -1.7101497932
## 781 782 783 784 785
## 0.0469844385 0.4207953999 2.0188929381 0.6684173227 0.5703197845
## 786 787 788 789 790
## 0.7198441690 3.1731734771 -0.7287289847 0.0469844385 -1.5839607546
## 791 792 793 794 795
## -0.4530155615 -0.4229281886 -1.1424586109 -1.6229281886 0.0537364655
## 796 797 798 799 800
## -1.0491172272 0.8761205805 2.0723156570 -2.9929342264 -2.3014953044
## 801 802 803 804 805
## 1.6099194663 2.5023096192 -1.9276843430 -1.2453123036 -0.1753062658
## 806 807 808 809 810
## 1.3294498887 3.0256449651 -0.8014953044 -1.6286355739 0.7032608500
## 811 812 813 814 815
## -2.2919829955 2.8294498887 3.6051633118 -4.7919829955 0.8294498887
## 816 817 818 819 820
## 0.8342060431 0.0818279659 -1.6191232650 1.8294498887 0.0537364655
## 821 822 823 824 825
## 0.6731734771 -0.1306314464 -0.7053936387 2.1498381312 1.2712710153
## 826 827 828 829 830
## -0.0744484456 0.7993625158 0.1217466308 0.4207953999 -0.5606254086
## 831 832 833 834 835
## -1.2006374842 0.4207953999 -0.1773021383 0.1122343218 2.1965088231
## 836 837 838 839 840
## -0.2053936387 1.5236490926 -1.2101497932 0.5703197845 1.6965088231
## 841 842 843 844 845
## 1.0188929381 -1.8782533692 -0.0044424078 0.1965088231 -0.7101497932
## 846 847 848 849 850
## -1.1072961005 2.2712710153 1.5469844385 -2.4811070619 -0.4625278704
## 851 852 853 854 855
## 3.0188929381 1.1217466308 0.9441307458 0.3226978617 -0.8268265229
## 856 857 858 859 860
## 3.1684173227 -0.4577717160 1.0469844385 3.5703197845 -2.0277777538
## 861 862 863 864 865
## -3.1820582928 0.1498381312 0.4207953999 2.0188929381 0.5703197845
## 866 867 868 869 870
## 3.5703197845 4.6051633118 1.5537364655 0.8994559265 1.0023096192
## 871 872 873 874 875
## -0.6025399460 -1.3268265229 -2.6353876009 0.6498381312 0.1265027853
## 876 877 878 879 880
## 0.2712710153 0.8741247080 1.6965088231 1.7198441690 -3.0511130997
## 881 882 883 884 885
## 0.9722222462 0.5236490926 -0.4391925245 1.0888989759 -3.6911251753
## 886 887 888 889 890
## -1.8830095237 0.7665148609 1.9722222462 -2.8220703684 -2.8268265229
## 891 892 893 894 895
## -3.4530155615 -2.1820582928 0.1450819767 -0.0792046001 0.7712710153
## 896 897 898 899 900
## -0.0744484456 0.9160392454 2.0188929381 -0.5372900627 1.0469844385
## 901 902 903 904 905
## -0.9996862533 -1.0558692542 1.1731734771 -1.1025399460 -2.1025399460
## 906 907 908 909 910
## -2.3130034859 1.6684173227 -1.9530155615 -1.9063448696 1.4955575922
## 911 912 913 914 915
## 0.6217466308 0.6965088231 0.9722222462 0.7946063613 1.2479356694
## 916 917 918 919 920
## 0.5003137467 0.4488869003 -0.6492106379 1.6450819767 0.7760271698
## 921 922 923 924 925
## 2.4722222462 -2.8549180233 -0.0044424078 1.9207953999 -0.8268265229
## 926 927 928 929 930
## -0.9811070619 -1.0091985623 0.1217466308 0.4160392454 -1.6306314464
## 931 932 933 934 935
## 1.2479356694 -0.6492106379 0.3460332076 0.9207953999 -1.5044424078
## 936 937 938 939 940
## 0.1684173227 -0.8315826773 -2.2053936387 1.0236490926 0.2198441690
## 941 942 943 944 945
## 0.6731734771 1.0188929381 0.4722222462 -2.4858632164 0.4488869003
## 946 947 948 949 950
## -1.7568204851 1.7246003235 -3.7006374842 0.0188929381 -1.3549180233
## 951 952 953 954 955
## -0.7568204851 1.4722222462 -0.0372900627 1.4955575922 -2.4344363700
## 956 957 958 959 960
## -1.3549180233 0.7198441690 0.2898502068 0.0469844385 -1.5044424078
## 961 962 963 964 965
## 0.5236490926 1.6450819767 0.1636611682 1.1684173227 -4.2520643306
## 966 967 968 969 970
## 1.0469844385 0.4908014377 -3.2753996765 1.5023096192 1.3808767350
## 971 972 973 974 975
## 0.5818279659 -1.3014953044 -2.2172208032 -2.2033977662 -3.2453123036
## 976 977 978 979
## -2.5910317646 1.2032608500 -0.8014953044 -2.6424586109Now that you can get at the residuals, we can use a similar method to extract the fitted (i.e., predicted) values. Use these two vectors to make a plot of residuals as a function of fitted values of the model m, to check for variation in the variance, within a call to plot(). Use the ~ notation as we did in the boxplot for your plot formula.
plot(m$residuals ~ m$fitted.values)
The residuals look to be ok, there is no great widening or narrowing of the distribution as we move along the x-axis. (There is one large residual up top, though). When assessing equal variance, we want to see a relatively homogenous cloud of points. We don’t want to see any structure in this cloud - the most common is when variance is greater at one end of your data’s range than another, giving a conical shape to your residuals.
For any categorical predictors in the model, we would make boxplots of the residuals conditional on these factors, as above.
For those who really want it, you can run a Bartlett’s test for equality of variances with the function `bartlett.test().
To resolve inhomogenous variance, you either have to transform the response variable, or use an approach that does not require homogeneity of variance (e.g., generalised least squares).
Are the data normally distributed?
Many statistical techniques assume that the data are normally distributed, including linear regression and t-tests. Violations of normality create problems for determining whether model coefficients are significantly different from zero and for calculating confidence intervals. Normality is not required for estimating the values of the coefficients themselves (although outliers may do …).
We can examine the normality of data by plotting a histogram. Plot a basic histogram of sparrow weight.
hist(sparrows$Weight)
The distribution is slightly skewed, with more lower values than higher values. For a test such as a t-test, all we can do to assess normality is plot the raw data. For a regression, we should again be checking the residuals. Plot a histogram of the residuals from the model ‘m’.
hist(m$residuals)
These look better! Less skewed.
Another graphical tool for examining normality is the normal probability plot or normal quantile plot of the residuals. We use the function qqnorm() to generate this plot. The residuals should fall more or less along a straight line. Make a qq plot of the residuals from the model m.
qqnorm(m$residuals)
It can be helpful to see a reference diagonal. Add the line by typing qqline(m$residuals).
qqnorm(m$residuals)
qqline(m$residuals)
A normal distribution is indicated by the points lying close to the diagonal reference line. A bow-shaped pattern of deviations from the diagonal indicates that the residuals have excessive skewness (i.e., they are not symmetrically distributed, with too many large errors in one direction). An S-shaped pattern of deviations indicates that the residuals have excessive kurtosis (i.e., there are either too many or two few large errors in both directions).
If you really need a p-value for your decision, you can use a Shapiro-Wilk test: `shapiro.test(). Run this test on the residuals of our model.
shapiro.test(m$residuals)##
## Shapiro-Wilk normality test
##
## data: m$residuals
## W = 0.98375, p-value = 5.661e-09The p-value in this case indicates that our distribution is significantly different from a normal distribution. Realistically, this will nearly always be the case, making the test somewhat unhelpful. Examining model fit through residuals will give you a better idea as to whether the distribution of your data is a problem.
For further reading on the issue of statistical tests of normality, see Zuur et al. 2010, Laasa 2009, and this link. As you may realise, real data with perfect Normal errors are extremely rare.
Are there lots of zeros in the data?
When working with count data (e.g., number of bugs on a leaf), it is common to have zeros. In some cases, one can have a lot of zeros. Such ‘zero-inflated’ data is problematical to analyze, and will require either a two-step approach first modelling the zeros and non-zeros using a binomial generalized linear model, and then only the non-zeros likely with a Poisson glm; or using a zero-inflated glm which essentially does these two steps at once. Much of this will make more sense when we start modeling associations in Unit 4.
Is there collinearity among the covariates?
Collinearity is the existence of correlation between covariates (the various predictor variables in your model). For example, weight and height are often tightly correlated, as are levels of soil nutrients. If collinear variables are all in the model, it is hard for the model to determine which variables are significant.
We can check for collinearity quickly with the plot() function again. Running plot() on a dataframe will generate a matrix of small plots, with each variable plotted against each other variable. The pairs() function does a similar thing. Run plot() on the sparrow dataset.
plot(sparrows)
You may need to expand the plotting window to interpret the output. Collinearity will show itself as an obvious relationship between two predictors. In this data set, nearly all of the anatomical variables are collinear, which makes sense. Because of this, you would need to take caution using more than one of these variables as predictors in a model, as they represent essentially the same thing (overall body size).
Are observations of the response variable independent?
A critical assumption of most statistical techniques is that observations are independent of one another. This assumption means that the value of any one data point is not influenced by the values of other data points, after the effects of the predictor variables have been taken into account.
What this means in practice, is that for data that we know are likely to be highly auto-correlated, such as time-series, repeated measures, or data with strong spatial structure such as tree growth or survival in a plot, we may or may not need to account for this structure in the model.
Auto-correlation refers to the tendency for observations close in space or time to be similar (i.e. to be correlated with one another).
It may be that the predictors we have in the model account for the auto-correlation and we do not need a explicit spatial or temporal model.
As with Normality and heterogeneity of variance, we can check the residuals of the model for evidence of autocorrelation in space or time (or phylogeny, or …).
Great, now you have some idea of how to explore data and test the assumptions of various statistical approaches. See the links on the Resources page for more background and more advanced techniques.
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