| title | output |
|---|---|
Basic Statistics with R |
github_document |
This lesson will cover calculating basic statistics with R, conducting statistical tests, and building simple linear models. We will use the 2007 NLA data for the examples.
First step in any project will be getting the data read into R. For this lesson we are using the 2007 National Lakes Assessment data, which ,luckily, we already have locally. From our nla_analysis.R script we can re-run a bunch of that code. I've copied over the important bits here. Alternatively, you can just open up your nla_analysis.R and run everything in there.
library(dplyr)
library(readr)
library(tidyr)
nla_wq_all <- read_csv("https://www.epa.gov/sites/production/files/2014-10/nla2007_chemical_conditionestimates_20091123.csv")
nla_wq <- nla_wq_all %>%
rename_all(tolower) %>% #making all names lower case beucase they are a mess!
mutate_if(is.character, tolower) %>%
filter(site_type == "prob_lake",
visit_no == 1) %>%
select(site_id, st, epa_reg, wsa_eco9, ptl, ntl, turb, chla, doc)Next, lets get a bit more info from the NLA Sites and join that to our data.
nla_sites <- read_csv("https://www.epa.gov/sites/production/files/2014-01/nla2007_sampledlakeinformation_20091113.csv")## Rows: 1252 Columns: 99
## ── Column specification ───────────────────────────────────────────────
## Delimiter: ","
## chr (78): SITE_ID, SAMPLED, DATE_COL, REPEAT, SITE_TYPE, LAKE_SAMP,...
## dbl (21): VISIT_NO, LON_DD, LAT_DD, ALBERS_X, ALBERS_Y, FLD_LON_DD,...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
nla_sites <- nla_sites %>%
filter(VISIT_NO == 1) %>%
select(SITE_ID, STATE_NAME, CNTYNAME, LAKE_ORIGIN, RT_NLA) %>%
rename_all(tolower) %>% #making all names lower case because they are a mess!
mutate_if(is.character, tolower)
nla <- left_join(nla_sites, nla_wq, by = "site_id") %>%
filter(!is.na(ntl),
!is.na(chla),
!is.na(ptl))So now we have our dataset ready for analysis.
First step in analyzing a dataset like this is going to be to dig through some basic statistics as well as some basic plots.
We can get a summary of the full data frame:
#Get a summary of the data frame
summary(nla)## site_id state_name cntyname
## Length:1028 Length:1028 Length:1028
## Class :character Class :character Class :character
## Mode :character Mode :character Mode :character
##
##
##
## lake_origin rt_nla st
## Length:1028 Length:1028 Length:1028
## Class :character Class :character Class :character
## Mode :character Mode :character Mode :character
##
##
##
## epa_reg wsa_eco9 ptl
## Length:1028 Length:1028 Min. : 1.0
## Class :character Class :character 1st Qu.: 11.0
## Mode :character Mode :character Median : 29.0
## Mean : 110.5
## 3rd Qu.: 94.5
## Max. :4679.0
## ntl turb chla
## Min. : 5.0 Min. : 0.237 Min. : 0.07
## 1st Qu.: 325.5 1st Qu.: 1.520 1st Qu.: 2.98
## Median : 586.5 Median : 3.815 Median : 7.79
## Mean : 1179.2 Mean : 13.620 Mean : 29.63
## 3rd Qu.: 1210.8 3rd Qu.: 11.200 3rd Qu.: 25.96
## Max. :26100.0 Max. :574.000 Max. :936.00
## doc
## Min. : 0.340
## 1st Qu.: 3.380
## Median : 5.575
## Mean : 8.863
## 3rd Qu.: 8.925
## Max. :290.570
Or, we can pick and choose what stats we want. For instance:
#Stats for Total Nitrogen
mean(nla$ntl)## [1] 1179.23
median(nla$ntl)## [1] 586.5
min(nla$ntl)## [1] 5
max(nla$ntl)## [1] 26100
sd(nla$ntl)## [1] 2086.885
IQR(nla$ntl)## [1] 885.25
range(nla$ntl)## [1] 5 26100
While visualization isn't the point of this lesson, some things are useful to do at this stage of analysis. In particular is looking at distributions and some basic scatterplots.
We can look at histograms and density:
#A single histogram using base
hist(nla$ntl)
#Log transform it
hist(log1p(nla$ntl)) #log1p adds one to deal with zeros
#Density plot
plot(density(log1p(nla$ntl)))
And boxplots:
#Simple boxplots
boxplot(nla$chla)
boxplot(log1p(nla$chla))
#Boxplots per group
boxplot(log1p(nla$chla)~nla$epa_reg)
And scatterplots:
#A single scatterplot
plot(log1p(nla$ptl),log1p(nla$chla))
And we can do a matrix of scatterplots from a data frame, but we should be careful as non-numeric columns won't log transform and many columns will make for a very uninformative matrix! So lets subset our dataframe and plot that.
# Subset with dplyr::select
nla_numeric_columns <- nla %>%
select(ntl, ptl, chla, turb)
#A matrix of scatterplot
plot(log1p(nla_numeric_columns))
There are way more tests than we can show examples for. For today we will show two very common and straightforward tests. The t-test and an ANOVA.
First we will look at the t-test to test and see if lake_orign shows a difference in chla. In other words can we expect a difference in clarity due to whether a lake is man-made or natural. This is a two-tailed test. There are two approaches for this 1) using the formula notation if your dataset is in a "long" format or 2) using two separate vectors if your dataset is in a "wide" format.
#Long Format - original format for lake_origin and chla
t.test(nla$chla ~ nla$lake_origin)##
## Welch Two Sample t-test
##
## data: nla$chla by nla$lake_origin
## t = -2.5178, df = 588.31, p-value = 0.01207
## alternative hypothesis: true difference in means between group man-made and group natural is not equal to 0
## 95 percent confidence interval:
## -21.428307 -2.647756
## sample estimates:
## mean in group man-made mean in group natural
## 24.48944 36.52747
#Wide Format - need to do some work to get there - tidyr is handy!
wide_nla <- nla %>%
select(site_id, chla, lake_origin) %>%
pivot_wider(names_from = lake_origin, values_from = chla)
names(wide_nla)<-c("site_id","man_made", "natural")
t.test(wide_nla$man_made, wide_nla$natural)##
## Welch Two Sample t-test
##
## data: wide_nla$man_made and wide_nla$natural
## t = 2.5178, df = 588.31, p-value = 0.01207
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 2.647756 21.428307
## sample estimates:
## mean of x mean of y
## 36.52747 24.48944
Same results, two different ways to approach. Take a look at the help (e.g. ?t.test) for more details on other types of t-tests (e.g. paired, one-tailed, etc.)
ANOVA can get involved quickly and I haven't done them since my last stats class, so I'm not the best to talk about these, but the very basics require fitting a model and wrapping that ins aov function. In the Getting More Help section I provide a link that would be a good first start for you ANOVA junkies. For todays lesson though, lets look at the simple case of a one-vay analysis of variance and check if reference class results in differences in our chlorophyll
# A quick visual of this:
boxplot(log1p(nla$chla)~nla$rt_nla)
# One way analysis of variance
nla_anova <- aov(log1p(chla)~rt_nla, data=nla)
nla_anova #Terms## Call:
## aov(formula = log1p(chla) ~ rt_nla, data = nla)
##
## Terms:
## rt_nla Residuals
## Sum of Squares 165.4115 1597.0869
## Deg. of Freedom 2 1025
##
## Residual standard error: 1.248252
## Estimated effects may be unbalanced
summary(nla_anova) #The table## Df Sum Sq Mean Sq F value Pr(>F)
## rt_nla 2 165.4 82.71 53.08 <2e-16 ***
## Residuals 1025 1597.1 1.56
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(nla_anova) #The table with a bit more## Analysis of Variance Table
##
## Response: log1p(chla)
## Df Sum Sq Mean Sq F value Pr(>F)
## rt_nla 2 165.41 82.706 53.08 < 2.2e-16 ***
## Residuals 1025 1597.09 1.558
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The last bit of basic stats we will cover is going to be linear relationships.
Let's first take a look at correlations. These can be done with cor().
#For a pair
cor(log1p(nla$ptl),log1p(nla$ntl))## [1] 0.8044302
#For a correlation matrix
cor(log1p(nla_numeric_columns))## ntl ptl chla turb
## ntl 1.0000000 0.8044302 0.7155305 0.6832722
## ptl 0.8044302 1.0000000 0.7132769 0.7933675
## chla 0.7155305 0.7132769 1.0000000 0.7254511
## turb 0.6832722 0.7933675 0.7254511 1.0000000
#Spearman Rank Correlations
cor(log1p(nla_numeric_columns),method = "spearman")## ntl ptl chla turb
## ntl 1.0000000 0.8156020 0.7030997 0.7077893
## ptl 0.8156020 1.0000000 0.7473225 0.8283843
## chla 0.7030997 0.7473225 1.0000000 0.7897786
## turb 0.7077893 0.8283843 0.7897786 1.0000000
You can also test for differences using:
cor.test(log1p(nla$ptl),log1p(nla$ntl))##
## Pearson's product-moment correlation
##
## data: log1p(nla$ptl) and log1p(nla$ntl)
## t = 43.375, df = 1026, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.7817373 0.8249955
## sample estimates:
## cor
## 0.8044302
Basic linear models in R can be built with the lm() function. If you aren't building standard least squares regressin models, (e.g. logistic) or aren't doing linear models then you will need to look elsewhere (e.g glm(), or nls()). For today our focus is going to be on simple linear models. Let's look at our ability to model chlorophyll, given the other variables we have.
# The simplest case
chla_tp <- lm(log1p(chla) ~ log1p(ptl), data=nla) #Creates the model
summary(chla_tp) #Basic Summary##
## Call:
## lm(formula = log1p(chla) ~ log1p(ptl), data = nla)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1358 -0.4810 -0.0091 0.5749 2.7828
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.17974 0.07445 2.414 0.0159 *
## log1p(ptl) 0.63363 0.01944 32.598 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9186 on 1026 degrees of freedom
## Multiple R-squared: 0.5088, Adjusted R-squared: 0.5083
## F-statistic: 1063 on 1 and 1026 DF, p-value: < 2.2e-16
names(chla_tp) #The bits## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"
chla_tp$coefficients #My preference## (Intercept) log1p(ptl)
## 0.1797429 0.6336289
coef(chla_tp) #Same thing, but from a function## (Intercept) log1p(ptl)
## 0.1797429 0.6336289
resid(chla_tp) # The residuals## 1 2 3 4 5
## -1.197616404 -0.890810064 0.310632312 -0.322657888 -0.406536743
## 6 7 8 9 10
## 0.127974131 -0.339407602 0.597741740 -0.016430236 -1.620567258
## 11 12 13 14 15
## 0.521617697 -1.178643606 -0.113580581 1.226329546 -2.801468058
## 16 17 18 19 20
## 0.700269818 -0.306449565 -0.475244893 1.359626277 -0.866209488
## 21 22 23 24 25
## 1.148038251 -0.561050897 -0.558791930 0.231390742 0.303802604
## 26 27 28 29 30
## 0.473821980 0.012330787 -0.770264263 0.219131943 0.681999289
## 31 32 33 34 35
## 0.333207658 0.837813121 -1.175729371 -0.375561135 -0.352459971
## 36 37 38 39 40
## 1.245037090 -0.573848002 -2.210292458 -2.013191415 -0.167865371
## 41 42 43 44 45
## -0.485736219 0.048888383 -0.387701994 1.295957132 -1.042656295
## 46 47 48 49 50
## -5.135798274 0.255540718 1.036477782 -0.897473109 0.009431088
## 51 52 53 54 55
## 0.981287264 -0.130470617 -1.115024047 -2.386592164 0.848151190
## 56 57 58 59 60
## 0.064101449 -0.075287557 1.348174257 0.310038814 -0.088312739
## 61 62 63 64 65
## 0.169516370 0.552235700 -0.662631754 0.141825723 -0.786121582
## 66 67 68 69 70
## 0.036541377 1.373140794 0.099982920 -0.460638133 -0.927829912
## 71 72 73 74 75
## -2.562620672 -0.095774787 1.313481002 -0.143509629 -2.553667039
## 76 77 78 79 80
## -0.957379017 0.164883222 -0.532164246 -0.251739859 -0.212270793
## 81 82 83 84 85
## 0.210521811 0.532133182 1.804047750 -0.638675511 0.274592940
## 86 87 88 89 90
## -0.920636682 -0.160163846 0.246426326 -0.313767389 0.001099233
## 91 92 93 94 95
## 1.205708026 -0.554252924 1.333263076 -0.625881185 2.178433214
## 96 97 98 99 100
## 0.330190161 -1.135994579 -0.679969824 1.515741002 -0.030232555
## 101 102 103 104 105
## 0.298845056 -0.274852948 0.108971214 -0.602551598 0.658975383
## 106 107 108 109 110
## -0.545627296 0.574612747 0.627682898 -0.220449921 0.564076030
## 111 112 113 114 115
## -0.155249873 0.137180990 -0.795774796 -1.241530103 0.412947599
## 116 117 118 119 120
## 0.212854474 -0.470352395 -0.047961443 1.002199289 0.121924408
## 121 122 123 124 125
## 0.791414435 0.027183033 -0.253387221 0.106996112 1.098067422
## 126 127 128 129 130
## -1.248116252 1.510433797 1.567930781 2.300381909 0.985681896
## 131 132 133 134 135
## 0.496089298 0.526329221 -0.837851435 -0.568257481 0.358103222
## 136 137 138 139 140
## -1.356275715 0.326527045 -0.141238047 0.602783859 -1.142627977
## 141 142 143 144 145
## -0.116055311 -1.170261637 -0.764239618 -0.590217717 -0.755585748
## 146 147 148 149 150
## -0.460600245 0.009374949 -0.456788384 0.493192664 0.258609346
## 151 152 153 154 155
## 0.468040685 0.035942213 0.146526852 -0.350930800 -0.752188031
## 156 157 158 159 160
## 0.054482770 0.337564959 -1.814229711 -0.405413830 -0.568541677
## 161 162 163 164 165
## 1.193876804 0.116799649 0.670181113 -0.029060467 0.125588431
## 166 167 168 169 170
## 0.284898525 -0.145672102 -1.798913546 -0.321531149 0.775083756
## 171 172 173 174 175
## 0.587029817 -1.753730134 0.340946257 -0.480983939 0.367852435
## 176 177 178 179 180
## 1.489149269 -0.049634422 0.555318601 0.558260667 0.123175096
## 181 182 183 184 185
## -0.273843610 -0.377016419 1.179086419 -0.499688648 2.204024320
## 186 187 188 189 190
## 1.795265026 -1.109988509 0.252619317 -0.138131188 -1.395725941
## 191 192 193 194 195
## -0.014162725 -1.805602472 0.195313188 -0.356079621 0.633850554
## 196 197 198 199 200
## 0.543096087 -0.213454984 0.869097732 -0.186882371 1.289382240
## 201 202 203 204 205
## -0.005546360 0.048095452 0.265388539 0.476196321 0.494601463
## 206 207 208 209 210
## 0.252626003 1.382084156 0.389016930 -0.549433637 -0.100147197
## 211 212 213 214 215
## 0.245500293 1.207810875 -1.200472397 -1.383265845 -1.486416853
## 216 217 218 219 220
## -1.034411361 -0.772960118 -0.765491997 1.594882039 -0.381750009
## 221 222 223 224 225
## 1.143462484 0.854380752 -0.258811878 0.282639016 -0.397527673
## 226 227 228 229 230
## -3.352116794 -0.038915340 1.556072666 -0.492848637 -0.415086666
## 231 232 233 234 235
## -0.481846131 -0.796512868 0.100799702 -0.344039900 -0.511520336
## 236 237 238 239 240
## 0.238001374 0.621111604 1.343813486 -1.614994467 -2.248785376
## 241 242 243 244 245
## -0.140803600 0.123978331 -0.265146545 -0.290403326 -0.374409356
## 246 247 248 249 250
## -0.976120812 -0.539864186 0.927095515 0.259208240 0.033462028
## 251 252 253 254 255
## 0.477299462 0.682319422 -1.077271621 -0.284075687 -0.651685110
## 256 257 258 259 260
## -0.344441220 0.057331346 0.902516069 -0.055092096 -0.133880159
## 261 262 263 264 265
## 0.377385463 1.339596706 0.518241212 0.085118123 0.129185443
## 266 267 268 269 270
## 0.411978825 2.044624374 -0.735404211 -3.730091487 -1.390589445
## 271 272 273 274 275
## 0.719518473 0.509275552 -0.924147769 -0.127210039 1.481114016
## 276 277 278 279 280
## -0.391800297 -0.397527673 0.860594041 -0.642965508 -0.242249150
## 281 282 283 284 285
## 0.083701308 -1.931216766 0.712235896 0.474573703 -0.285765536
## 286 287 288 289 290
## 0.735847520 -0.270859521 -1.181859784 0.017583463 -0.548280725
## 291 292 293 294 295
## 0.922980237 0.053121488 -0.506223954 -0.708320163 -0.564495884
## 296 297 298 299 300
## 0.218306535 0.160293509 0.322648992 0.095404276 0.345072173
## 301 302 303 304 305
## -1.184025199 -0.513797731 0.139063918 0.763318336 -0.133796576
## 306 307 308 309 310
## -0.303116201 -1.528676829 -0.662381726 -0.496965440 -1.011338680
## 311 312 313 314 315
## 1.569201847 -1.873334728 0.455097561 -2.085842294 0.603338659
## 316 317 318 319 320
## 1.259204517 0.526505129 0.977339183 0.069193649 0.008269120
## 321 322 323 324 325
## 0.898956954 -0.016675728 0.790173699 1.588889635 0.164140589
## 326 327 328 329 330
## 0.367265049 -0.774460667 -0.906100536 -1.297297949 0.219782011
## 331 332 333 334 335
## -0.123621551 -0.288549758 0.608837882 0.930545452 -0.136514841
## 336 337 338 339 340
## -0.321550798 -1.338846952 -0.285493837 0.963593860 1.176333305
## 341 342 343 344 345
## -0.228972404 -0.008895207 0.616364349 0.481613487 -0.104012962
## 346 347 348 349 350
## -0.126586842 0.831227195 0.150991104 0.492366850 -0.563637373
## 351 352 353 354 355
## -0.801877450 -0.210991310 1.126386904 -0.426202808 1.362539444
## 356 357 358 359 360
## 0.633821979 -1.326509310 0.584712042 -0.355041549 -0.073423963
## 361 362 363 364 365
## -0.468651150 0.721449444 -0.148072952 -1.578302037 -0.664082431
## 366 367 368 369 370
## -0.211984481 0.498471217 -0.299367908 -0.170737164 -1.367059762
## 371 372 373 374 375
## 1.053411892 -0.005334372 -0.063566165 -0.494458756 -0.522777490
## 376 377 378 379 380
## 0.315597975 1.765546925 -2.958741787 -0.518426183 -0.046452619
## 381 382 383 384 385
## -0.324698742 -0.259961233 -0.496437052 1.367202294 -0.655330779
## 386 387 388 389 390
## -0.965285649 -0.551840377 -0.064887287 0.635133244 1.485938467
## 391 392 393 394 395
## 0.019503704 0.911453715 0.345847185 -0.292671218 0.130174924
## 396 397 398 399 400
## 1.423428688 1.316788851 2.202364283 1.184110357 0.465833723
## 401 402 403 404 405
## 0.182049896 1.020862884 0.819890620 -0.256674243 -0.107360255
## 406 407 408 409 410
## -1.008687601 -0.231771964 0.507705442 -0.043384121 -0.481055241
## 411 412 413 414 415
## 0.087657867 0.809051653 -0.674282399 -0.335433077 -0.562994098
## 416 417 418 419 420
## -0.216048153 -0.058501105 -1.498575058 0.123322373 -0.782621464
## 421 422 423 424 425
## 1.104981684 -0.463134613 1.249244596 0.253962501 0.822198510
## 426 427 428 429 430
## 0.959791409 -0.372407533 -0.188243565 0.682355547 0.340247212
## 431 432 433 434 435
## 1.053617928 0.747879252 -0.366620135 -0.036725370 0.009315156
## 436 437 438 439 440
## -1.375032223 0.877552547 -0.911918595 -0.299402267 0.974523816
## 441 442 443 444 445
## 1.279177323 0.425412713 -0.884002930 -0.417002480 -0.027219246
## 446 447 448 449 450
## 0.243121352 0.109178978 -1.106909194 -0.234738327 0.265343180
## 451 452 453 454 455
## 0.570095053 1.332971639 0.622256282 -0.053306784 -2.178738631
## 456 457 458 459 460
## 1.561665053 -1.014123656 -0.923822620 0.383820581 -0.009235935
## 461 462 463 464 465
## 0.653803441 -0.083922242 0.463987581 0.263773803 -1.584264377
## 466 467 468 469 470
## -0.993005566 -0.094919139 0.172529911 0.178787270 -0.948458606
## 471 472 473 474 475
## 0.138797192 0.166697569 1.066726726 -0.976417085 -0.014625023
## 476 477 478 479 480
## -0.208464234 -0.870566793 0.543781698 -0.970647102 0.018346451
## 481 482 483 484 485
## 1.523087756 -0.998493037 0.547203646 0.084156521 -0.295311107
## 486 487 488 489 490
## -0.731700817 -0.955634471 -0.366620135 -1.082338734 -0.551282341
## 491 492 493 494 495
## 0.513908565 0.123937695 0.674927803 -0.194852074 -0.124165897
## 496 497 498 499 500
## -0.880301075 0.414469214 -1.108208375 -0.369394601 -0.421395835
## 501 502 503 504 505
## 0.212163241 -0.219537165 -0.537259046 -1.764016920 0.546123540
## 506 507 508 509 510
## -0.437562572 0.668193509 -1.589352293 -0.213712464 -1.304450454
## 511 512 513 514 515
## 0.983150391 -0.287798187 0.323960897 0.330432777 1.138455098
## 516 517 518 519 520
## 1.451143506 0.526876178 -0.193673255 0.240983674 -0.429766982
## 521 522 523 524 525
## -0.283238526 -0.381945616 -3.181131493 -0.232194975 0.640271868
## 526 527 528 529 530
## -0.186795239 -0.027363539 -0.780818923 0.665237200 -0.445124428
## 531 532 533 534 535
## 0.443936431 1.267988864 0.239132889 -0.320624594 0.753718172
## 536 537 538 539 540
## -0.041913415 -0.223130161 2.148050987 0.645674710 0.203383890
## 541 542 543 544 545
## -0.499402343 1.866534173 -2.046797133 0.246368117 -0.597362970
## 546 547 548 549 550
## 1.302714941 -0.395519658 1.483795059 0.110784684 -0.479179048
## 551 552 553 554 555
## -0.031154325 0.305291205 1.390276344 -1.212562363 -0.765469446
## 556 557 558 559 560
## 0.138219953 1.610329834 -0.353055811 -0.373942731 -0.859018055
## 561 562 563 564 565
## -0.422750116 0.174157916 -0.210991310 1.707429429 -0.068127147
## 566 567 568 569 570
## -1.128301947 -0.448125965 0.751606468 0.337207558 0.509540404
## 571 572 573 574 575
## -0.212241655 -0.387829583 -0.808677774 -1.133700802 0.295688441
## 576 577 578 579 580
## -0.859402451 -0.366360877 -0.133495599 0.371564047 1.353750182
## 581 582 583 584 585
## 0.125370673 -0.193344714 -0.420759663 0.150940030 -0.629240121
## 586 587 588 589 590
## 0.231515021 -0.187721120 -0.574746198 -0.834995509 0.161591526
## 591 592 593 594 595
## -0.165870874 -0.142815617 0.012624591 0.146526852 -0.028579436
## 596 597 598 599 600
## -0.615220588 0.186765103 0.124889699 0.628123159 0.650853454
## 601 602 603 604 605
## 2.575348764 2.782760375 -0.024954577 1.854042546 -0.584415298
## 606 607 608 609 610
## 0.913160323 0.544049031 0.598112568 -0.346743647 1.331802040
## 611 612 613 614 615
## 0.099285401 -0.629884114 -0.501488563 1.459732097 -0.144935038
## 616 617 618 619 620
## -1.018732471 0.293194665 0.587912425 0.777002752 0.628644453
## 621 622 623 624 625
## 0.402477114 0.233036833 0.089647952 -0.946234674 0.421335722
## 626 627 628 629 630
## 1.232743942 0.076956576 0.280855241 -0.393274978 1.028318697
## 631 632 633 634 635
## 0.964649698 0.670355245 1.512079969 -0.833477455 -0.297852150
## 636 637 638 639 640
## -0.249936087 0.441505460 0.667763188 0.293903420 -3.855912697
## 641 642 643 644 645
## 0.787738451 0.456617326 -0.981023755 0.365688826 1.751849592
## 646 647 648 649 650
## -0.134818521 -0.398023876 -0.075747975 -0.415627714 0.995394044
## 651 652 653 654 655
## -0.531151918 0.823941336 0.078225940 -0.007572652 -0.056720882
## 656 657 658 659 660
## -0.363395586 -2.271845254 -0.359542016 0.422399119 1.481805098
## 661 662 663 664 665
## -0.755212712 1.157333750 -1.233498802 -1.534367313 -0.323156861
## 666 667 668 669 670
## -0.207988131 -0.111042769 0.863865936 -1.853345954 1.513747010
## 671 672 673 674 675
## -0.240528356 -0.303249576 0.092432968 -0.001610298 -0.477805937
## 676 677 678 679 680
## 0.983705600 0.119829084 0.261861259 -0.359980232 0.402151051
## 681 682 683 684 685
## 0.427027154 1.145744461 1.066531849 -0.001897477 1.560050548
## 686 687 688 689 690
## 0.338712620 0.765503343 0.325457626 0.972507176 1.217142228
## 691 692 693 694 695
## 0.269089178 -0.946029432 -0.811241679 0.653325412 0.150997302
## 696 697 698 699 700
## 0.976731135 -0.050981852 0.788654338 -0.266363302 1.431422819
## 701 702 703 704 705
## -0.036640800 1.310610096 -0.422334707 1.671848054 0.429440458
## 706 707 708 709 710
## -0.402063743 -0.509314006 1.790558751 0.204695283 0.142649388
## 711 712 713 714 715
## -1.226801964 -0.118105272 -0.329255377 -0.364871450 -0.339401361
## 716 717 718 719 720
## -2.307338409 -0.622915894 -0.045277772 -0.327937887 -1.903900374
## 721 722 723 724 725
## -0.202240170 -0.469033811 0.323303732 -0.727109091 0.626561763
## 726 727 728 729 730
## 1.022350296 -0.470027784 0.659057915 -0.659362940 0.235296305
## 731 732 733 734 735
## -0.092345283 -0.185635614 -0.048184765 1.350841487 0.895851602
## 736 737 738 739 740
## 1.180302198 0.662679025 -0.647628559 2.187773624 -1.277977695
## 741 742 743 744 745
## -0.903914888 -2.102542290 0.119550032 0.103131909 1.244231358
## 746 747 748 749 750
## -1.221930091 1.726768798 -0.778713824 1.002394481 -0.415086666
## 751 752 753 754 755
## 0.667533036 0.447551807 -0.223191130 -0.190686044 0.785902973
## 756 757 758 759 760
## -0.254555341 -0.603767988 -0.778722284 -0.666826614 0.852477799
## 761 762 763 764 765
## -0.572464726 0.905206495 0.546559387 -0.303475408 0.029105187
## 766 767 768 769 770
## -0.315609935 -0.121093105 -1.255820904 -0.253665017 1.719196690
## 771 772 773 774 775
## -0.175159980 -0.387196409 0.657100044 -0.652673952 -0.495467471
## 776 777 778 779 780
## -0.062611241 0.403126937 0.358288213 0.083898444 -1.153666497
## 781 782 783 784 785
## 0.643081639 -1.205713614 0.148398867 -1.594807435 0.112862574
## 786 787 788 789 790
## 0.966173618 0.288377317 1.810024642 0.936208863 2.683622115
## 791 792 793 794 795
## -1.250399220 -0.586157470 -0.156114700 1.151351158 -1.896956423
## 796 797 798 799 800
## 0.230772312 -0.887037834 -0.306578702 0.949790828 -0.212763001
## 801 802 803 804 805
## -0.115681130 -0.133495599 -0.008741538 -0.426867283 1.715134510
## 806 807 808 809 810
## -3.489212101 1.300017238 2.518521375 1.358623414 0.961398342
## 811 812 813 814 815
## -0.324854993 0.483491771 1.526484973 -0.022574121 -0.486579450
## 816 817 818 819 820
## -0.690021801 0.254453414 -0.170784889 -0.083698626 0.262625763
## 821 822 823 824 825
## -0.018684876 0.519291136 0.235474338 -0.348069615 0.225352803
## 826 827 828 829 830
## 1.127345096 -0.204874852 0.057517715 0.604661844 -0.111702252
## 831 832 833 834 835
## -0.142856191 -0.108328425 -0.978543337 1.594939550 0.521929751
## 836 837 838 839 840
## 0.239228713 -0.869691880 1.087755693 0.496069187 0.701858199
## 841 842 843 844 845
## -0.240421985 -2.812818495 0.191989226 1.107229086 -1.991804595
## 846 847 848 849 850
## 1.738975014 -0.075287557 1.067984934 0.164268412 0.095933512
## 851 852 853 854 855
## -0.240504554 -0.750757711 0.854169950 0.757754547 -1.728215750
## 856 857 858 859 860
## -2.096161228 -0.764239618 -1.036430637 0.661252376 -1.467474710
## 861 862 863 864 865
## 0.447206152 -1.015232794 0.382907010 -1.186294984 -1.192618493
## 866 867 868 869 870
## -0.389146913 0.870291278 -0.257003918 -0.551321458 0.007454227
## 871 872 873 874 875
## 1.505446223 -0.388670452 1.531765919 -0.730223506 0.508537232
## 876 877 878 879 880
## -1.059126090 -0.066002439 -0.450228286 2.081613182 -0.431285921
## 881 882 883 884 885
## 0.330470936 1.722981129 1.230283025 -0.230371508 0.768583978
## 886 887 888 889 890
## -0.491652728 0.651661179 0.357513405 1.161339150 0.624179451
## 891 892 893 894 895
## -1.287303490 -1.226083734 0.864933699 1.580694196 0.358718743
## 896 897 898 899 900
## 0.874122069 0.774360404 -0.171204324 0.566929452 0.877212800
## 901 902 903 904 905
## 0.683469871 0.281220360 -0.582279545 -1.233023411 -0.186845694
## 906 907 908 909 910
## -0.287876575 0.661245337 -0.958209078 -0.459486476 -0.306361309
## 911 912 913 914 915
## 0.654835954 -0.882605348 -1.223623281 0.427093256 -0.344574703
## 916 917 918 919 920
## -1.175142363 0.007124826 -0.008937546 0.381494854 -0.856220066
## 921 922 923 924 925
## -1.105535468 1.322958859 -0.540290508 0.367507505 -0.954174152
## 926 927 928 929 930
## -0.921201362 0.781230594 -0.635141499 -0.400353097 -2.541054565
## 931 932 933 934 935
## -0.416285174 -2.038116839 -0.429516129 -0.464938385 -1.354954370
## 936 937 938 939 940
## -1.799895412 0.599425410 0.262187913 1.231560213 0.197873551
## 941 942 943 944 945
## -0.067388308 1.344725044 0.013938663 0.053211806 1.120638421
## 946 947 948 949 950
## 0.493944946 -1.200221544 0.616988657 -1.531770249 0.299973300
## 951 952 953 954 955
## 1.659965531 -0.743363234 -2.052433653 -0.264776869 0.831782460
## 956 957 958 959 960
## 0.156749916 0.013107875 0.737592844 -0.718524930 -0.535728585
## 961 962 963 964 965
## -0.708435197 -0.495619484 -0.887311240 0.230948255 -0.192406532
## 966 967 968 969 970
## -1.114058765 -0.527539240 -0.411040230 0.855861327 -0.295991983
## 971 972 973 974 975
## 0.506775802 -0.573765738 -1.334738143 0.017635839 -3.190505253
## 976 977 978 979 980
## 0.532166291 -0.844611900 0.761080799 0.451253437 0.762383722
## 981 982 983 984 985
## 1.095739212 0.448936112 -0.398639678 0.761907674 -0.050418147
## 986 987 988 989 990
## -0.003868449 0.374840768 0.130524111 1.035497850 0.297349742
## 991 992 993 994 995
## -0.405696926 1.960286761 0.880488364 -0.643302182 -0.478489810
## 996 997 998 999 1000
## 0.623150583 -0.381272956 -0.024646382 -0.151556102 0.662002400
## [ reached getOption("max.print") -- omitted 28 entries ]
We can also do multiple linear regression.
chla_tp_tn_turb <- lm(log1p(chla) ~ log1p(ptl) + log1p(ntl) + log1p(turb), data = nla)
summary(chla_tp_tn_turb)##
## Call:
## lm(formula = log1p(chla) ~ log1p(ptl) + log1p(ntl) + log1p(turb),
## data = nla)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7115 -0.4347 0.0188 0.5247 2.2272
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.60380 0.19192 -8.357 < 2e-16 ***
## log1p(ptl) 0.10869 0.03480 3.123 0.00184 **
## log1p(ntl) 0.43622 0.04045 10.783 < 2e-16 ***
## log1p(turb) 0.45109 0.03708 12.167 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8081 on 1024 degrees of freedom
## Multiple R-squared: 0.6206, Adjusted R-squared: 0.6194
## F-statistic: 558.2 on 3 and 1024 DF, p-value: < 2.2e-16
There's a lot more we can do with linear models including dummy variables (character or factors will work), interactions, etc. That's a bit more than we want to get into. Again the link below is a good place to start for more info.
One nice site that covers basic stats in R is Quick R: Basic Statistics. There are others, but that is a good first stop.