Objectives for chapter 06
SwR Achievements
Resource 6.1 : Data, codebook, and R packages for learning about t-test
Data
Two options for accessing the data:
nhanes_2015–2016_ch6.csv from https://edge.sagepub.com/harris1e.Codebook
Two options for accessing the codebook:
nhanes_demographics_20152016_codebook.html and nhanes_examination_20152016_codebook.html from https://edge.sagepub.com/harris1e.Packages
Example 6.1 : Numbered Example Title
R Code 6.1 : Get NHANES data for blood pressure examination with demographics variable for 2015-2016
## run one once manually #########
## list EXAM tables for 2016 to get file names
exam_tables_2016 <- nhanesA::nhanesTables('EXAM', 2016)
## list variables in BPX_I (Blood Pressure file)
bpx_i_variables <- nhanesA::nhanesTableVars('EXAM', 'BPX_I')
bpx_i <- nhanesA::nhanes('BPX_I')
demo_i <- nhanesA::nhanes('DEMO_I')
bpx_2016 <- dplyr::full_join(demo_i, bpx_i, by = "SEQN")
save_data_file("chap06", bpx_2016, "bpx_2016.rds")(For this R code chunk is no output available. For the raw data see )
R Code 6.2 : Get NHANES codebook for blood pressure examination with demographics variable for 2015-2016
cb_systolic <- nhanesA::nhanesCodebook("BPX_I", "BPXSY1")
cb_diastolic <- nhanesA::nhanesCodebook("BPX_I", "BPXDI1")(For this R code chunk is no output available. For the raw data see)
Besides to call the appropriate website for 2015-2016 examination codebook and demographic codebook there is also the option to download information via {nhanesA}.
Example 6.2 : Numbered Example Title
R Code 6.3 : Glance at NHANES data for blood pressure examination with demographics variable for 2015-2016
| Name | bpx_2016 |
| Number of rows | 9971 |
| Number of columns | 67 |
| _______________________ | |
| Column type frequency: | |
| factor | 45 |
| numeric | 22 |
| ________________________ | |
| Group variables | None |
Variable type: factor
| skim_variable | n_missing | complete_rate | ordered | n_unique | top_counts |
|---|---|---|---|---|---|
| RIDSTATR | 0 | 1.00 | FALSE | 2 | Bot: 9544, Int: 427 |
| RIAGENDR | 0 | 1.00 | FALSE | 2 | Fem: 5079, Mal: 4892 |
| RIDRETH1 | 0 | 1.00 | FALSE | 5 | Non: 3066, Non: 2129, Mex: 1921, Oth: 1547 |
| RIDRETH3 | 0 | 1.00 | FALSE | 6 | Non: 3066, Non: 2129, Mex: 1921, Oth: 1308 |
| RIDEXMON | 427 | 0.96 | FALSE | 2 | May: 4950, Nov: 4594 |
| DMQMILIZ | 3822 | 0.62 | FALSE | 2 | No: 5622, Yes: 527 |
| DMQADFC | 9444 | 0.05 | FALSE | 3 | No: 267, Yes: 258, Ref: 2 |
| DMDBORN4 | 0 | 1.00 | FALSE | 3 | Bor: 7733, Oth: 2236, Don: 2 |
| DMDCITZN | 2 | 1.00 | FALSE | 4 | Cit: 8785, Not: 1168, Ref: 9, Don: 7 |
| DMDYRSUS | 7735 | 0.22 | FALSE | 11 | 20 : 384, 10 : 285, 30 : 281, 1 y: 273 |
| DMDEDUC3 | 7324 | 0.27 | FALSE | 19 | 1st: 252, Nev: 239, 2nd: 237, 3rd: 220 |
| DMDEDUC2 | 4252 | 0.57 | FALSE | 6 | Som: 1692, Col: 1422, Hig: 1236, Les: 688 |
| DMDMARTL | 4252 | 0.57 | FALSE | 8 | Mar: 2886, Nev: 1048, Div: 614, Liv: 555 |
| RIDEXPRG | 8683 | 0.13 | FALSE | 3 | The: 1125, Can: 93, Yes: 70 |
| SIALANG | 0 | 1.00 | FALSE | 2 | Eng: 8584, Spa: 1387 |
| SIAPROXY | 1 | 1.00 | FALSE | 2 | No: 6281, Yes: 3689 |
| SIAINTRP | 0 | 1.00 | FALSE | 2 | No: 9514, Yes: 457 |
| FIALANG | 329 | 0.97 | FALSE | 2 | Eng: 8430, Spa: 1212 |
| FIAPROXY | 329 | 0.97 | FALSE | 2 | No: 9633, Yes: 9 |
| FIAINTRP | 329 | 0.97 | FALSE | 2 | No: 9237, Yes: 405 |
| MIALANG | 2994 | 0.70 | FALSE | 2 | Eng: 6382, Spa: 595 |
| MIAPROXY | 2993 | 0.70 | FALSE | 2 | No: 6921, Yes: 57 |
| MIAINTRP | 2993 | 0.70 | FALSE | 2 | No: 6632, Yes: 346 |
| AIALANGA | 4009 | 0.60 | FALSE | 3 | Eng: 5218, Spa: 638, Asi: 106 |
| DMDHHSIZ | 0 | 1.00 | FALSE | 7 | 4: 2061, 2: 1723, 3: 1719, 5: 1672 |
| DMDFMSIZ | 0 | 1.00 | FALSE | 7 | 4: 2011, 5: 1635, 3: 1634, 2: 1510 |
| DMDHHSZA | 0 | 1.00 | FALSE | 4 | 0: 6298, 1: 2147, 2: 1199, 3 o: 327 |
| DMDHHSZB | 0 | 1.00 | FALSE | 5 | 0: 4715, 1: 1990, 2: 1833, 3: 822 |
| DMDHHSZE | 0 | 1.00 | FALSE | 4 | 0: 7151, 1: 1663, 2: 1099, 3 o: 58 |
| DMDHRGND | 0 | 1.00 | FALSE | 2 | Mal: 5053, Fem: 4918 |
| DMDHRBR4 | 396 | 0.96 | FALSE | 4 | Bor: 6359, Oth: 3207, Ref: 5, Don: 4 |
| DMDHREDU | 396 | 0.96 | FALSE | 6 | Som: 2908, Col: 2331, Hig: 2015, 9-1: 1200 |
| DMDHRMAR | 62 | 0.99 | FALSE | 8 | Mar: 5681, Nev: 1305, Liv: 1017, Div: 977 |
| DMDHSEDU | 4745 | 0.52 | FALSE | 7 | Col: 1629, Som: 1462, Hig: 980, Les: 619 |
| INDHHIN2 | 345 | 0.97 | FALSE | 16 | $10: 1634, $25: 1017, $35: 960, $75: 920 |
| INDFMIN2 | 329 | 0.97 | FALSE | 16 | $10: 1548, $25: 1038, $35: 934, $75: 885 |
| PEASCCT1 | 9735 | 0.02 | FALSE | 3 | Tim: 178, Oth: 46, SP : 12 |
| BPAARM | 2573 | 0.74 | FALSE | 3 | Rig: 7361, Lef: 36, Cou: 1 |
| BPACSZ | 2585 | 0.74 | FALSE | 5 | Lar: 3368, Adu: 2461, Thi: 1068, Chi: 488 |
| BPXPULS | 657 | 0.93 | FALSE | 2 | Reg: 9113, Irr: 201 |
| BPXPTY | 2595 | 0.74 | FALSE | 2 | Rad: 7348, Bra: 28 |
| BPAEN1 | 2826 | 0.72 | FALSE | 2 | No: 7142, Yes: 3 |
| BPAEN2 | 2658 | 0.73 | FALSE | 2 | No: 7130, Yes: 183 |
| BPAEN3 | 2695 | 0.73 | FALSE | 2 | No: 7070, Yes: 206 |
| BPAEN4 | 9647 | 0.03 | FALSE | 2 | No: 248, Yes: 76 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| SEQN | 0 | 1.00 | 88717.00 | 2878.52 | 83732.00 | 86224.50 | 88717.00 | 91209.50 | 93702.0 | ▇▇▇▇▇ |
| SDDSRVYR | 0 | 1.00 | 9.00 | 0.00 | 9.00 | 9.00 | 9.00 | 9.00 | 9.0 | ▁▁▇▁▁ |
| RIDAGEYR | 0 | 1.00 | 31.90 | 24.77 | 0.00 | 9.00 | 27.00 | 53.00 | 80.0 | ▇▃▃▃▃ |
| RIDAGEMN | 9276 | 0.07 | 10.79 | 7.02 | 0.00 | 5.00 | 10.00 | 17.00 | 24.0 | ▇▇▆▆▅ |
| RIDEXAGM | 5911 | 0.41 | 104.53 | 68.97 | 0.00 | 41.00 | 100.00 | 162.00 | 239.0 | ▇▆▆▅▅ |
| DMDHRAGE | 0 | 1.00 | 46.18 | 15.83 | 18.00 | 34.00 | 44.00 | 57.00 | 80.0 | ▅▇▆▃▃ |
| WTINT2YR | 0 | 1.00 | 31740.15 | 32929.54 | 3293.93 | 12878.50 | 20160.47 | 33257.36 | 233755.8 | ▇▁▁▁▁ |
| WTMEC2YR | 0 | 1.00 | 31740.15 | 34105.57 | 0.00 | 12550.53 | 20281.32 | 33708.15 | 242386.7 | ▇▁▁▁▁ |
| SDMVPSU | 0 | 1.00 | 1.49 | 0.50 | 1.00 | 1.00 | 1.00 | 2.00 | 2.0 | ▇▁▁▁▇ |
| SDMVSTRA | 0 | 1.00 | 126.27 | 4.24 | 119.00 | 123.00 | 126.00 | 130.00 | 133.0 | ▇▇▇▇▇ |
| INDFMPIR | 1052 | 0.89 | 2.27 | 1.58 | 0.00 | 0.97 | 1.82 | 3.48 | 5.0 | ▇▇▅▃▆ |
| BPXCHR | 8033 | 0.19 | 106.56 | 21.76 | 58.00 | 90.00 | 104.00 | 120.00 | 190.0 | ▃▇▅▂▁ |
| BPXPLS | 2595 | 0.74 | 74.61 | 12.23 | 36.00 | 66.00 | 74.00 | 82.00 | 142.0 | ▁▇▃▁▁ |
| BPXML1 | 2600 | 0.74 | 146.60 | 18.56 | 110.00 | 130.00 | 140.00 | 160.00 | 260.0 | ▇▅▁▁▁ |
| BPXSY1 | 2826 | 0.72 | 120.54 | 18.62 | 72.00 | 108.00 | 118.00 | 130.00 | 236.0 | ▂▇▂▁▁ |
| BPXDI1 | 2826 | 0.72 | 66.18 | 14.29 | 0.00 | 58.00 | 66.00 | 76.00 | 120.0 | ▁▁▇▅▁ |
| BPXSY2 | 2658 | 0.73 | 120.32 | 18.62 | 76.00 | 108.00 | 118.00 | 130.00 | 238.0 | ▃▇▂▁▁ |
| BPXDI2 | 2658 | 0.73 | 66.06 | 14.37 | 0.00 | 58.00 | 66.00 | 76.00 | 144.0 | ▁▂▇▁▁ |
| BPXSY3 | 2695 | 0.73 | 119.95 | 18.29 | 76.00 | 108.00 | 116.00 | 130.00 | 226.0 | ▃▇▂▁▁ |
| BPXDI3 | 2695 | 0.73 | 65.99 | 14.56 | 0.00 | 58.00 | 66.00 | 76.00 | 140.0 | ▁▂▇▁▁ |
| BPXSY4 | 9647 | 0.03 | 129.10 | 22.88 | 82.00 | 112.00 | 128.00 | 142.50 | 212.0 | ▃▇▅▁▁ |
| BPXDI4 | 9647 | 0.03 | 70.38 | 18.62 | 0.00 | 62.00 | 72.00 | 80.00 | 132.0 | ▁▁▇▃▁ |
my_glance_data(bpx_2016)#> obs SEQN SDDSRVYR RIDSTATR RIAGENDR RIDAGEYR
#> 1 1 83732 9 Both interviewed and MEC examined Male 62
#> 2 356 84087 9 Both interviewed and MEC examined Male 36
#> 3 1252 84983 9 Interviewed only Female 10
#> 4 2369 86100 9 Both interviewed and MEC examined Male 33
#> 5 3954 87685 9 Both interviewed and MEC examined Female 1
#> 6 5273 89004 9 Both interviewed and MEC examined Female 0
#> 7 7700 91431 9 Interviewed only Female 47
#> 8 8826 92557 9 Both interviewed and MEC examined Male 71
#> 9 9290 93021 9 Both interviewed and MEC examined Female 18
#> 10 9971 93702 9 Both interviewed and MEC examined Female 24
#> RIDAGEMN RIDRETH1 RIDRETH3
#> 1 NA Non-Hispanic White Non-Hispanic White
#> 2 NA Other Race - Including Multi-Racial Non-Hispanic Asian
#> 3 NA Non-Hispanic Black Non-Hispanic Black
#> 4 NA Mexican American Mexican American
#> 5 22 Other Race - Including Multi-Racial Non-Hispanic Asian
#> 6 3 Mexican American Mexican American
#> 7 NA Other Hispanic Other Hispanic
#> 8 NA Other Race - Including Multi-Racial Non-Hispanic Asian
#> 9 NA Mexican American Mexican American
#> 10 NA Non-Hispanic White Non-Hispanic White
#> RIDEXMON RIDEXAGM DMQMILIZ DMQADFC
#> 1 November 1 through April 30 NA No <NA>
#> 2 November 1 through April 30 NA No <NA>
#> 3 <NA> NA <NA> <NA>
#> 4 November 1 through April 30 NA No <NA>
#> 5 May 1 through October 31 23 <NA> <NA>
#> 6 November 1 through April 30 3 <NA> <NA>
#> 7 <NA> NA No <NA>
#> 8 May 1 through October 31 NA No <NA>
#> 9 November 1 through April 30 226 No <NA>
#> 10 May 1 through October 31 NA No <NA>
#> DMDBORN4 DMDCITZN
#> 1 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 2 Others Not a citizen of the US
#> 3 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 4 Others Not a citizen of the US
#> 5 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 6 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 7 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 8 Others Citizen by birth or naturalization
#> 9 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> 10 Born in 50 US states or Washington, DC Citizen by birth or naturalization
#> DMDYRSUS DMDEDUC3
#> 1 <NA> <NA>
#> 2 15 year or more, but less than 20 years <NA>
#> 3 <NA> 3rd grade
#> 4 10 year or more, but less than 15 years <NA>
#> 5 <NA> <NA>
#> 6 <NA> <NA>
#> 7 <NA> <NA>
#> 8 1 year or more, but less than 5 years <NA>
#> 9 <NA> High school graduate
#> 10 <NA> <NA>
#> DMDEDUC2 DMDMARTL
#> 1 College graduate or above Married
#> 2 College graduate or above Living with partner
#> 3 <NA> <NA>
#> 4 9-11th grade (Includes 12th grade with no diploma) Married
#> 5 <NA> <NA>
#> 6 <NA> <NA>
#> 7 College graduate or above Separated
#> 8 9-11th grade (Includes 12th grade with no diploma) Married
#> 9 <NA> <NA>
#> 10 College graduate or above Never married
#> RIDEXPRG SIALANG SIAPROXY SIAINTRP FIALANG
#> 1 <NA> English No No English
#> 2 <NA> English No No English
#> 3 <NA> English Yes No English
#> 4 <NA> Spanish No No Spanish
#> 5 <NA> English Yes Yes English
#> 6 <NA> English Yes No English
#> 7 <NA> English No No English
#> 8 <NA> English No Yes <NA>
#> 9 <NA> English No No English
#> 10 The participant was not pregnant at exam English No No English
#> FIAPROXY FIAINTRP MIALANG MIAPROXY MIAINTRP AIALANGA
#> 1 No No English No No English
#> 2 No No English No No English
#> 3 No No <NA> <NA> <NA> <NA>
#> 4 No No English No No Spanish
#> 5 No Yes <NA> <NA> <NA> <NA>
#> 6 No No <NA> <NA> <NA> <NA>
#> 7 No No <NA> <NA> <NA> <NA>
#> 8 <NA> <NA> English No Yes <NA>
#> 9 No No English No No English
#> 10 No No English No No English
#> DMDHHSIZ DMDFMSIZ DMDHHSZA
#> 1 2 2 0
#> 2 4 4 0
#> 3 2 2 0
#> 4 6 6 1
#> 5 5 5 3 or more
#> 6 7 or more people in the Household 7 or more people in the Family 2
#> 7 4 4 1
#> 8 3 3 0
#> 9 4 1 0
#> 10 3 1 0
#> DMDHHSZB DMDHHSZE DMDHRGND DMDHRAGE DMDHRBR4
#> 1 0 1 Male 62 Born in 50 US states or Washington, DC
#> 2 2 0 Male 36 Others
#> 3 1 0 Female 35 Born in 50 US states or Washington, DC
#> 4 3 0 Female 36 Others
#> 5 0 0 Male 37 Others
#> 6 1 0 Male 37 Others
#> 7 0 0 Female 23 Born in 50 US states or Washington, DC
#> 8 0 1 Male 71 Others
#> 9 0 0 Female 26 <NA>
#> 10 0 0 Female 22 Born in 50 US states or Washington, DC
#> DMDHREDU DMDHRMAR
#> 1 College Graduate or above Married
#> 2 College Graduate or above Living with partner
#> 3 Some College or AA degree Never married
#> 4 9-11th Grade (Includes 12th grade with no diploma) Married
#> 5 High School Grad/GED or Equivalent Married
#> 6 Less Than 9th Grade Married
#> 7 High School Grad/GED or Equivalent Living with partner
#> 8 9-11th Grade (Includes 12th grade with no diploma) Married
#> 9 <NA> Never married
#> 10 College Graduate or above Never married
#> DMDHSEDU WTINT2YR WTMEC2YR
#> 1 High School Grad/GED or Equivalent 134671.370 135629.507
#> 2 <NA> 18448.501 18550.188
#> 3 <NA> 14840.527 0.000
#> 4 9-11th Grade (Includes 12th grade with no diploma) 26689.378 27745.104
#> 5 High School Grad/GED or Equivalent 8860.151 8793.912
#> 6 Less Than 9th Grade 5454.854 5610.739
#> 7 <NA> 19288.330 0.000
#> 8 <NA> 19926.979 25066.422
#> 9 <NA> 16127.645 16401.812
#> 10 <NA> 107361.907 105080.445
#> SDMVPSU SDMVSTRA INDHHIN2 INDFMIN2 INDFMPIR PEASCCT1
#> 1 1 125 $65,000 to $74,999 $65,000 to $74,999 4.39 <NA>
#> 2 1 133 $100,000 and Over $100,000 and Over 5.00 <NA>
#> 3 2 132 $15,000 to $19,999 $15,000 to $19,999 0.94 <NA>
#> 4 1 128 $15,000 to $19,999 $15,000 to $19,999 0.48 <NA>
#> 5 1 122 $15,000 to $19,999 $15,000 to $19,999 0.63 <NA>
#> 6 2 121 $55,000 to $64,999 $55,000 to $64,999 1.50 <NA>
#> 7 1 121 $35,000 to $44,999 $35,000 to $44,999 1.65 <NA>
#> 8 1 129 <NA> <NA> NA <NA>
#> 9 2 128 $45,000 to $54,999 $ 5,000 to $ 9,999 0.59 <NA>
#> 10 2 119 $65,000 to $74,999 $35,000 to $44,999 3.54 <NA>
#> BPXCHR BPAARM BPACSZ BPXPLS BPXPULS BPXPTY BPXML1 BPXSY1 BPXDI1
#> 1 NA Right Large (15X32) 76 Regular Radial 150 128 70
#> 2 NA Right Adult (12X22) 46 Regular Radial 160 122 80
#> 3 NA <NA> <NA> NA <NA> <NA> NA NA NA
#> 4 NA Right Large (15X32) 68 Regular Radial 150 116 70
#> 5 110 <NA> <NA> NA Regular <NA> NA NA NA
#> 6 122 <NA> <NA> NA Regular <NA> NA NA NA
#> 7 NA <NA> <NA> NA <NA> <NA> NA NA NA
#> 8 NA Right Adult (12X22) 72 Regular Radial 140 112 80
#> 9 NA Right Child (9X17) 74 Regular Radial 120 100 74
#> 10 NA Right Adult (12X22) 80 Regular Radial 150 118 66
#> BPAEN1 BPXSY2 BPXDI2 BPAEN2 BPXSY3 BPXDI3 BPAEN3 BPXSY4 BPXDI4 BPAEN4
#> 1 No 124 64 No 116 62 No NA NA <NA>
#> 2 No 128 82 No 122 86 No NA NA <NA>
#> 3 <NA> NA NA <NA> NA NA <NA> NA NA <NA>
#> 4 No 124 66 No 116 66 No NA NA <NA>
#> 5 <NA> NA NA <NA> NA NA <NA> NA NA <NA>
#> 6 <NA> NA NA <NA> NA NA <NA> NA NA <NA>
#> 7 <NA> NA NA <NA> NA NA <NA> NA NA <NA>
#> 8 No 118 78 No 122 80 No NA NA <NA>
#> 9 No 100 66 No 98 68 No NA NA <NA>
#> 10 No 114 68 No 124 64 No NA NA <NA>R Code 6.4 : Glance at NHANES codebook for systolic & diastolic blood pressure (2015-2016)
#> *********************** Systolic blood pressure ******************
#> $`Variable Name:`
#> [1] "BPXSY1"
#>
#> $`SAS Label:`
#> [1] "Systolic: Blood pres (1st rdg) mm Hg"
#>
#> $`English Text:`
#> [1] "Systolic: Blood pressure (first reading) mm Hg"
#>
#> $`Target:`
#> [1] "Both males and females 8 YEARS -\r 150 YEARS"
#>
#> $BPXSY1
#> # A tibble: 2 × 5
#> `Code or Value` `Value Description` Count Cumulative `Skip to Item`
#> <chr> <chr> <int> <int> <lgl>
#> 1 72 to 236 Range of Values 7145 7145 NA
#> 2 . Missing 2399 9544 NA
#>
#>
#> *********************** Diastolic blood pressure ******************
#> $`Variable Name:`
#> [1] "BPXDI1"
#>
#> $`SAS Label:`
#> [1] "Diastolic: Blood pres (1st rdg) mm Hg"
#>
#> $`English Text:`
#> [1] "Diastolic: Blood pressure (first reading) mm Hg"
#>
#> $`Target:`
#> [1] "Both males and females 8 YEARS -\r 150 YEARS"
#>
#> $BPXDI1
#> # A tibble: 2 × 5
#> `Code or Value` `Value Description` Count Cumulative `Skip to Item`
#> <chr> <chr> <int> <int> <lgl>
#> 1 0 to 120 Range of Values 7145 7145 NA
#> 2 . Missing 2399 9544 NAR Code 6.5 : Clean NHANES blood pressure data (2015-2016)
## load bpx_2016 #######
bpx_2016 <- base::readRDS("data/chap06/bpx_2016.rds")
bp_clean <- bpx_2016 |>
dplyr::rename(
systolic = BPXSY1,
systolic2 = BPXSY2,
sex = RIAGENDR
) |>
dplyr::mutate(diff_syst = systolic - systolic2) |>
dplyr::relocate(c(systolic, systolic2, diff_syst), .before = sex)
save_data_file("chap06", bp_clean, "bp_clean.rds")(For this R code chunk is no output available)
For this first achievement we are going to look into the relationship between one categorical variable and one continuous variable using histograms, means, and standard deviations.
Example 6.3 : Description of blood pressure data from NHANES 2015-2016
R Code 6.6 : Histogram of systolic blood pressure (NHANES 2015-2016)
## load bpx_2016 #######
bpx_2016 <- base::readRDS("data/chap06/bpx_2016.rds")
## graph systolic blood pressure variable BPXSY1 (Figure 6.1)
sys_histo <- bpx_2016 |>
ggplot2::ggplot(
ggplot2::aes(x = BPXSY1)
) +
ggplot2::geom_histogram(
fill = "mediumpurple",
color = "white",
bins = 30,
na.rm = TRUE
) +
ggplot2::theme_bw() +
ggplot2::labs(
x = "Systolic blood pressure (mmHg)",
y = "NHANES participants"
)
sys_histo
This is the replication of the book’s Figure 6.1.
The graph is not exactly normally distributed; it has a little right skew. The quantile values (0% 25% 50% 75% 100%) are 72, 108, 118, 130, 236. The middle 50% lies in the range between 108 and 130 mmHG. You can’t see the highest values because their frequencies are too small.
R Code 6.7 : Histogram of systolic blood pressure with risk factors (NHANES 2015-2016)
## graph systolic blood pressure with risk factors (Figure 6.2)
sys_histo2 <- bpx_2016 |>
ggplot2::ggplot(
ggplot2::aes(
x = BPXSY1,
fill = BPXSY1 > 120)
) +
ggplot2::geom_histogram(
color = "white",
bins = 30,
na.rm = TRUE) +
ggplot2::theme_bw() +
ggplot2::labs(
x = "Systolic blood pressure (mmHg)",
y = "NHANES participants"
) +
ggplot2::scale_fill_manual(
values = c("mediumpurple", "grey"),
labels = c("Normal range",
"at-risk or high"),
name = "Systolic\nBlood Pressure"
)
sys_histo2
This is the replication of the book’s Figure 6.2.
R Code 6.8 : Blood pressure histogram with several colors according to their medical conditions
## graph systolic blood pressure differentiated
sys_histo3 <- bpx_2016 |>
dplyr::select(BPXSY1) |>
dplyr::mutate(sys = dplyr::case_when(
BPXSY1 < 105 ~ "0",
BPXSY1 >= 105 & BPXSY1 < 120 ~ "1",
BPXSY1 >= 120 & BPXSY1 < 130 ~ "2",
BPXSY1 >= 130 & BPXSY1 < 140 ~ "3",
BPXSY1 >= 140 ~ "4"
)
) |>
ggplot2::ggplot(
ggplot2::aes(x = BPXSY1, fill = sys)
) +
ggplot2::geom_histogram(
color = "white",
binwidth = 2,
na.rm = TRUE) +
ggplot2::theme_bw() +
ggplot2::theme(legend.position = "bottom") +
ggplot2::labs(
x = "Systolic blood pressure (mmHg)",
y = "NHANES participants"
) +
ggplot2::scale_fill_manual(
values = c(
"0" = "grey",
"1" = "mediumpurple",
"2" = "yellow",
"3" = "darkorange",
"4" = "red"
),
labels = c("Low",
"Optimal",
"Normal",
"At-risk",
"High"
),
name = "Systolic\nBlood Pressure"
) +
ggplot2::xlim(70, 240)
sys_histo3
Here I have experimented to colorize the histogram with different colors. I took as borders the medical condition for isolated blood pressure measures:
In this case I can’t use the color directly as fill variable into the ggplots::aes() function. Besides I learned two other solve two other issues:
breaks or to provide borders conforming to the medical status. Only binwidth of 1 and 2 worked, 3 already showed the problem. Other people had the same problem, see for instance the section “Example 2: Draw Histogram with Different Colors Using ggplot2 Package” in Draw Histogram with Different Colors in R (2 Examples) (Schork, n.d.).R Code 6.9 : Histogram of diastolic blood pressure with risk factors (NHANES 2015-2016)
## graph systolic blood pressure with risk factors (Figure 6.3)
dia_histo <- bpx_2016 |>
ggplot2::ggplot(
ggplot2::aes(
x = BPXDI1,
fill = BPXDI1 > 80)
) +
ggplot2::geom_histogram(
color = "white",
bins = 30,
na.rm = TRUE) +
ggplot2::theme_bw() +
ggplot2::labs(
x = "Diastolic blood pressure (mmHg)",
y = "NHANES participants"
) +
ggplot2::scale_fill_manual(
values = c("mediumpurple", "grey"),
labels = c("Normal range",
"at-risk or high"),
name = "Diastolic\nBlood Pressure"
)
dia_histo
This is the replication of the book’s Figure 6.3.
We have a mean of the systolic blood pressure a little bit above 120.54 mmHG. This is almost exactly the upper cutoff value of 120 mmHG for the “normal range” of blood pressure. Is this only valid for the sample of also for the whole population? Have about half of the US people high systolic blood pressure, e.g. more than 120mmHG? The question can be answered with a one-sample t-test. The one sample t-test compares a sample mean to a hypothesized or population mean.
There are three different t-tests
The t-distribution has a bell shape like the normal distribution. But unlike the normal distribution its variance is not known but approximated with its only parameter degrees of freedom (df). df is calculated by the number of observations minus one (\(n-1\)). With higher degrees of freedom the t-distribution will get closer to the normal distribution. Often the number 30 is recommended as the cutting point where t-distribution and normal distribution are equivalent.
I am following Procedure 5.2 from Section 5.8.
Write the null and alternate hypotheses:
Two considerations:
Compute the test statistic. The one-sample t-test uses the t-statistic (sort of like a z-statistic)
Formula 6.1 : t-test formula
\[ t = \frac{m_{x} - \mu_{x}}{\frac{s_x}{\sqrt{n_{x}}}} \tag{6.1}\]
The formula is very similar as the Z-score statistic in Equation 4.1. The only difference is that in the above t-statistic the denominator is the standard deviation rather than the standard error.
z shows how many sample standard deviations some value is away from the mean.t shows how many standard errors (i.e., population standard deviations) some value is away from the mean.\[ t = \frac{120.5394 - 120}{\frac{18.61692}{\sqrt{7145}}} = 2.45 \]
R Code 6.11 : t-statistic of systolic blood pressure aof NHANES sample with hypothesized population mean
(
t_test_systolic <- stats::t.test(
bpx_2016$BPXSY1,
alternative = "two.sided",
mu = 120)
)#>
#> One Sample t-test
#>
#> data: bpx_2016$BPXSY1
#> t = 2.4491, df = 7144, p-value = 0.01435
#> alternative hypothesis: true mean is not equal to 120
#> 95 percent confidence interval:
#> 120.1077 120.9711
#> sample estimates:
#> mean of x
#> 120.5394## for later use #########
t_sys = t_test_systolic[["statistic"]][["t"]]
df_sys = t_test_systolic[["parameter"]][["df"]]
null_sys = t_test_systolic[["null.value"]][["mean"]]
estimate_sys = t_test_systolic[["estimate"]][["mean of x"]]
p_value_sys = t_test_systolic[["p.value"]]
se_sys = t_test_systolic[["stderr"]]I have stored the result of the t-test into variables for later use.
Assessment 6.1 : Explications of the t-test output
1. Line: Data (variable) used. 2. Line: - t: value of the t-test statistic. - df: degrees of freedom, with t-statistic = subtracting 1 from sample size = \(n - 1\). - p-value: The probability of the sample coming from a population where the null hypothesis is true. 3. Line: wording of the alternative hypothesis. 4. Line: Chosen confidence interval. 5. Line: The lower and upper boundary of the confidence interval. 6. Line: Sample estimates that has to be compared to the value of the null hypothesis.
Review and interpret the test statistics: Calculate the probability that your test statistic is at least as big as it is if there is no relationship (i.e., the null is true).
The following examples replicates Figure 6.4, 6.5 and 6.6. I will break down the final code into several steps following the nice article [Visualizing Sampling Distributions: Learn how to add areas under the curve in sampling distributions]https://ggplot2tutor.com/tutorials/sampling_distributions (Burkhart 2021).
Example 6.4 : Probability distribution of t-test statistic
R Code 6.12 : Student t distributions with 1 degree of freedom (df)
Procedure 6.1 : Plotting a distribution with {ggplot2}
ggplot2::stat_function draws the function. We can specify the function extra or create an anonymous function or — as I have done here — use a function from an R package. Note that there is no parenthesis behind the function name.R Code 6.13 : Student t distributions with 1 and 7144 degree of freedom (df) and normal distribution compared
ggplot2::ggplot(tibble::tibble(x = c(-7, 7)),
ggplot2::aes(x)) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = 1),
geom = "line",
linewidth = 0.7,
ggplot2::aes(linetype = "1")
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = 7144),
geom = "line",
linewidth = 0.7,
ggplot2::aes(linetype = "5")
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = 30),
geom = "line",
linewidth = 0.7,
ggplot2::aes(linetype = "3")
) +
ggplot2::stat_function(
fun = stats::dnorm,
geom = "line",
linewidth = 0.7,
ggplot2::aes(color = "red")
) +
ggplot2::scale_linetype_discrete(
name = "t dist",
labels = c("df = 1", "df = 30", "df = 7144")
) +
ggplot2::scale_color_discrete(
name = "normal dist",
labels = "mean = 0, sd = 1"
) +
ggplot2::theme_bw()
The plot shows two things:
Procedure 6.2 : Plotting several distributions with {ggplot2}
ggplot2::stat_function().ggplot2::aes() function.R Code 6.14 : t-distribution (df = 7,144) shaded for values of 2.4491 or higher (replicating Figure 6.5)
ggplot2::ggplot() +
ggplot2::xlim(-4, 4) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "line",
linewidth = 0.7
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(t_sys, 4),
ggplot2::aes(fill =
paste("t >=", round(t_sys, 3))
)
) +
ggplot2::theme_bw() +
ggplot2::scale_fill_manual(
name = "",
values = "steelblue"
)
The plot shows that the t-value of 2.499 is very unlikely if the null hypotheses were true, e.g. if the sample comes from a population with a systolic blodd pressure mean of 240.
R Code 6.15 : t-distribution (df = 7,144) with 2.5% shaded in each tail of the distribution (replicating Figure 6.6)
ggplot2::ggplot() +
ggplot2::xlim(-4, 4) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "line",
linewidth = 0.7
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(1.96, 4),
ggplot2::aes(fill =
paste("Rejection region")
)
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(-4, -1.96),
ggplot2::aes(fill =
paste("Rejection region")
)
) +
ggplot2::theme_bw() +
ggplot2::scale_fill_manual(
name = "",
values = "purple3"
) +
ggplot2::labs(
x = "t-statistic",
y = "Probability density"
)
The plot shows the rejection regions for the probability of 95%.
\[ \begin{align*} p_{low}(x > 1.96) \approx 0.025 \\ p_{high}(x < 1.96) \approx 0.975 \\ p_{high} - p_{low} = \\ 0.975 - 0.025 = 0.95 \end{align*} \]
R Code 6.16 : t-distribution (df = 7,144) with 2.5% shaded in each tail of the distribution (replicating Figure 6.6)
ggplot2::ggplot() +
ggplot2::xlim(-4, 4) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "line",
linewidth = 0.7
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(1.96, 4),
alpha = 0.5,
ggplot2::aes(fill =
paste("Rejection region"),
)
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(-4, -1.96),
alpha = 0.5,
ggplot2::aes(fill =
paste("Rejection region"),
),
) +
ggplot2::stat_function(
fun = stats::dt,
args = list(df = df_sys),
geom = "area",
xlim = c(t_sys, 4),
alpha = 0.5,
ggplot2::aes(fill =
paste("t >=", round(t_sys, 3))
)
) +
ggplot2::theme_bw() +
ggplot2::scale_fill_manual(
name = "",
values = c("purple3", "red")
) +
ggplot2::labs(
x = "t-statistic",
y = "Probability density"
)
The plot shows that the t-value is in the rejection area, e.g., the null has to be rejected.
Conclude and write the report.
Even though the difference between the mean systolic blood pressure of 120.54 and the hypothesized value of 120 is small, it is statistically significant. The probability of this sample that it comes from a population where the mean systolic blood pressure is actually 120 is just 1.4%. This sample is likely to be from a population with a higher mean blood pressure.
The mean systolic blood pressure in a sample of 7145 people was 120.54 (sd = 18.62). A one-sample t-test found this mean to be statistically significantly different from the hypothesized mean of 120 [t(7144) = 2.449; p = 0.014]. The sample likely came from a population with a mean systolic blood pressure not equal to 120.
Instead of comparing one mean to a hypothesized or population mean, the independent-samples t-test compares the means of two groups to each other.
We could for instance be interested to see if the blood pressure for persons of different sex are the same. Or the question statistically formulated: Do males and females in the sample come from a population where males and females have the same mean systolic blood pressure?
I am not going into the details of achievement 3 because there is no much difference between the procedure for the one-sample t-test and the independent-samples t-test. Essentially there are only two differences:
Formula 6.2 : Independent-samples t-test formula
\[ t = \frac{m_{1} - m_{2}}{\sqrt{\frac{s_1^2}{n_{1}} + \frac{s_2^2}{n_{2}}}} \tag{6.2}\]
R Code 6.17 : Independent-samples t-test for systolic blood pressure of males and females
bp_clean = base::readRDS("data/chap06/bp_clean.rds")
bp_clean |>
tidyr::drop_na(systolic) |>
dplyr::group_by(sex) |>
dplyr::summarize(
mean_systolic = mean(systolic),
var_systolic = var(systolic),
sample_size = dplyr::n()
)
two_sample_t <- t.test(formula =
bp_clean$systolic ~ bp_clean$sex)
two_sample_t#> # A tibble: 2 × 4
#> sex mean_systolic var_systolic sample_size
#> <fct> <dbl> <dbl> <int>
#> 1 Male 122. 329. 3498
#> 2 Female 119. 358. 3647
#>
#> Welch Two Sample t-test
#>
#> data: bp_clean$systolic by bp_clean$sex
#> t = 7.3135, df = 7143, p-value = 2.886e-13
#> alternative hypothesis: true difference in means between group Male and group Female is not equal to 0
#> 95 percent confidence interval:
#> 2.347882 4.067432
#> sample estimates:
#> mean in group Male mean in group Female
#> 122.1767 118.9690It is important to note that category variables like sex do not work with the default x, y version of the t-test. Therefore we have to apply the method with class formula.
In R formulae a single variable on the left-hand side is followed by the tilde sign ~ and one ore more objects that predict or explain the left-hand side.
In a lot of statistical tests, the object on the left-hand side of the formula is the outcome or dependent variable while the object(s) on the right-hand side of the formula are the predictors or independent variables. (SwR)
It is commonly used to generate design matrices for modeling function (e.g.
lm) (Kuhn 2017).
In our case, systolic blood pressure is the outcome being explained by the predictor of sex.
In R the default t-test for independent samples is the Welch’s t-test and not the student t-test.
Welch’s t-test is slightly different from the original formula for t, which used pooled variance in the denominator. Pooled variance assumes that the variances in the two groups are equal and combines them. (SwR)
There is an scientific article explaining why Welch’s t-test should be used in any case, even if the assumption of homogeneity of variance is met:
We show that the Welch’s t-test provides a better control of Type 1 error rates when the assumption of homogeneity of variance is not met, and it loses little robustness compared to Student’s t-test when the assumptions are met. We argue that Welch’s t-test should be used as a default strategy. (Delacre, Lakens, and Leys 2017; see also: Delacre, Lakens, and Leys 2022)
Just to conclude this abbreviated section I quote the final summary reporting the independent t-test results.
There was a statistically significant difference [t(7143) = 7.31; p < .05] in mean systolic blood pressure between males (m = 122.18) and females (m = 118.97) in the sample. The sample was taken from the U.S. population, indicating that males in the United States likely have a different mean systolic blood pressure than females in the United States. The difference between male and female mean systolic blood pressure was 3.21 in the sample; in the population this sample came from, the difference between male and female mean blood pressure was likely to be between 2.35 and 4.07 (d = 3.21; 95% CI: 2.35–4.07). (SwR)
Again: I am not going to summarize this section because it resembles achievement 2 (one-sample t-test) and achievement 3 (independent.samples t-test).
Formula 6.3 : Independent-samples t-test formula
\[ t = \frac{m_{d} - 0}{\sqrt{\frac{s_d}{n_{d}}}} \tag{6.3}\]
General advice before starting a test statistics
Always look at some visuals and descriptive statistics before you are starting the test procedure and following the NHST procedure as outlined in Procedure 5.2.
This shows that the difference of the mean between two measures is very small (0.55 mmHG). But it turned out that this value is highly statistically significant. But from a clinical point of view it is irrelevant!
Statistically significant != meaningful!
All our three t-tests result in small but statistically significant values. This is an important reminder that statistically significant p-values are not necessarily of relevance.
By the way: The reason for our small but statistically significant values are very large samples.
The computation in R is the same as in Listing / Output 6.2. Again apply the formula version of Welch’s t-test with the only difference to add the argument paired = TRUE.
We haven seen that even the very small difference of 0.54 mmHG systolic blood pressure is with a large sample size statistically significant. But this small difference is clinically not relevant. To judge the importance of some statistically significant results we need effect sizes as another criteria.
The proportion of people that believe that effect sizes are even more important than p-values is rising. P-values only report whether a difference or relationship from a sample is likely to be true in the population, while effect sizes provide information about the strength or size of a difference or relationship.
In Chapter 5 we discussed for the Chi-squared test as effect sizes
For t-test the appropriate effect size measure is Cohen’s d.
Resource 6.2 Delving into effect size assessment
Derived from the importance of effect size parameter I need to know more how to use and interpret different kind of effect sizes. What follows are some resources
Formula 6.4 : Formula for Cohen’s d one-sample t-test
\[ d = \frac{m_{x} - \mu_{x}}{s_{x}} \tag{6.4}\]
\(m_{x}\) = sample mean for \(x\) \(\mu_{x}\) = hypothesized or population mean \(s_{x}\) = sample standard deviation for \(x\)
The formula is similar to the already well-known z-score calculation (Equation 4.1).
Assessment 6.2 : Classification of Cohen’s d values
Resource 6.3 Packages with Cohen’s d functions
Thee is another package {effsize} that I have not used. I had problems to use it, because it has no argument for \(\mu\) but it also has not many downloads.
The packages {effectsize} and {rstatix} are important as they have many other computation for effect size parameters.
#> # A tibble: 4 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 rstatix 5551 2024-03-21 2024-03-27
#> 2 effectsize 1479 2024-03-21 2024-03-27
#> 3 lsr 452 2024-03-21 2024-03-27
#> 4 effsize 289 2024-03-21 2024-03-27Example 6.5 : Computation of Cohen’s d for one-sample t-tests
R Code 6.18 : Computation of Cohen’s d for one-sample t-tests manually (by hand)
#> [1] 0.02897354The effect size is very small, it is not even close the starting value for “small effect size”.
The mean systolic blood pressure in a sample of 7,145 people was 120.54 (sd = 18.62). A one-sample t-test found this mean to be statistically significantly different from the hypothesized mean of 120 [t(7144) = 2.45; p = 0.014]. The sample likely came from a population with a mean systolic blood pressure not equal to 120. While the sample mean was statistically significantly different from 120, is has a very small effect size (Cohen’s d = .03).
R Code 6.19 : Compute Cohen’s d with for one-sample t-tests {lsr}
lsr::cohensD(bp_clean$systolic, mu = 120)#> [1] 0.02897354R Code 6.20 : Compute Cohen’s d for one-sample t-tests with {effectsize}
effectsize::cohens_d(
x = bp_clean$systolic,
mu = 120
)#> Warning: Missing values detected. NAs dropped.#> Cohen's d | 95% CI
#> ------------------------
#> 0.03 | [0.01, 0.05]
#>
#> - Deviation from a difference of 120.The {effectsize} packages discusses two alternatives for Cohen’s d:
R Code 6.21 : Compute Cohen’s d for one-sample t-tests with {rstatix}
rstatix::cohens_d(
data = bp_clean,
formula = systolic ~ 1,
mu = 120
)#> # A tibble: 1 × 6
#> .y. group1 group2 effsize n magnitude
#> * <chr> <chr> <chr> <dbl> <int> <ord>
#> 1 systolic 1 null model 0.0290 7145 negligibleFormula 6.5 : Formula for independent-samples cohen’s d
\[ d = \frac{m_{1}-{m_{2}}}{\sqrt{\frac{s_{1}^2 + s_{2}^2}{2}}} \tag{6.5}\]
Example 6.6 : Computation of Cohen’s d for independent-samples t-tests
R Code 6.22 : Computation of Cohen’s d for independent-samples t-tests manually (by hand)
bp_clean <- base::readRDS("data/chap06/bp_clean.rds")
bp_ind_samples <-
bp_clean |>
tidyr::drop_na(systolic) |>
dplyr::group_by(sex) |>
dplyr::summarize(
mean_sys_sex = mean(systolic),
var_sys_sex = var(systolic)
)
bp_ind_samples
m1 <- bp_ind_samples[[1,2]]
m2 <- bp_ind_samples[[2,2]]
var1 <- bp_ind_samples[[1,3]]
var2 <- bp_ind_samples[[2,3]]
(m1 - m2) / (sqrt((var1 + var2) / 2))#> # A tibble: 2 × 3
#> sex mean_sys_sex var_sys_sex
#> <fct> <dbl> <dbl>
#> 1 Male 122. 329.
#> 2 Female 119. 358.
#> [1] 0.1730045The effect size is very small.
R Code 6.23 : Compute Cohen’s d of independent samples with {lsr}
lsr::cohensD(
x = systolic ~ sex,
data = bp_clean,
method = "unequal"
)#> [1] 0.1730045R Code 6.24 : Compute Cohen’s d for independent-samples t-tests with {effectsize}
effectsize::cohens_d(
x = bp_clean$systolic,
y = bp_clean$sex,
pooled_sd = FALSE,
paired = FALSE
)#> Warning: Missing values detected. NAs dropped.#> Cohen's d | 95% CI
#> ------------------------
#> 0.17 | [0.13, 0.22]
#>
#> - Estimated using un-pooled SD.The {effectsize} packages discusses two alternatives for Cohen’s d:
R Code 6.25 : Compute Cohen’s d for independent-samples t-tests with {rstatix}
rstatix::cohens_d(
data = bp_clean,
formula = systolic ~ sex
)#> # A tibble: 1 × 7
#> .y. group1 group2 effsize n1 n2 magnitude
#> * <chr> <chr> <chr> <dbl> <int> <int> <ord>
#> 1 systolic Male Female 0.173 3498 3647 negligiblepaired = FALSE because we have an independent samples t-test.var.equal = FALSE because we have used the Welch’s t-test taht does not assume equal variances.Formula 6.6 : Formula for dependent-samples cohen’s d
\[ d = \frac{m_{d}-0}{s_{d}} \tag{6.6}\]
Example 6.7 : Computation of Cohen’s d for dependent-samples t-tests
R Code 6.26 : Computation of Cohen’s d for dependent-samples t-tests manually (by hand)
#> mean_diff sd_diff
#> 1 0.5449937 4.898043
#> [1] 0.1112676The effect size is very small.
R Code 6.27 : Compute Cohen’s d of dependent samples with {lsr}
lsr::cohensD(
x = bp_clean$systolic,
y = bp_clean$systolic2,
method = "paired")#> [1] 0.1112676Instead of the default method (“pooled”) we need “paired” as method for the dependent-samples t-test.
R Code 6.28 : Compute Cohen’s d for dependent-samples t-tests with {effectsize}
effectsize::cohens_d(
x = bp_clean$systolic,
y = bp_clean$systolic2,
paired = TRUE
)#> Warning: Missing values detected. NAs dropped.#> For paired samples, 'repeated_measures_d()' provides more options.#> Cohen's d | 95% CI
#> ------------------------
#> 0.11 | [0.09, 0.13]The {effectsize} packages discusses two alternatives for Cohen’s d:
R Code 6.29 : Compute Cohen’s d for dependent-samples t-tests with {rstatix}
bp_clean |>
dplyr::select(systolic, systolic2) |>
tidyr::drop_na() |>
tidyr::pivot_longer(
cols = c("systolic", "systolic2"),
names_to = "treatment",
values_to = "value") |>
rstatix::cohens_d(
formula = value ~ treatment,
paired = TRUE
)#> # A tibble: 1 × 7
#> .y. group1 group2 effsize n1 n2 magnitude
#> * <chr> <chr> <chr> <dbl> <int> <int> <ord>
#> 1 value systolic systolic2 0.111 7101 7101 negligibleIn order to get a working cohen’s d computation with {rstatix} I had to apply tidyr::pivot_longer().
In this section I am not following the sequence of the book. I start with an overview and will separate - checking visually the normality assumption - testing with different approaches the normality assumption
Bullet List
Example 6.8 : One-sample t-test: Checking the normality assumption visually
R Code 6.30 : Checking normality visually (30 bins)
The data appear right-skewed, e.g. the right tail is longer than the left tail. There is also the assumption that the distribution is bimodal, e.g., has two modes.
The best way to check if this assumption is correct is to change the number of bins.
R Code 6.31 : Checking normality visually (60 bins)
my_hist_dnorm(
df = bp_clean2,
v = bp_clean2$systolic,
n_bins = 60,
x_label = "Systolic blood pressure (mmHg)"
)
Another graph — this time with 60 bins — shows that we have a unimodal distribution.
R Code 6.32 : Checking normality visually with a Q-Q plot
bp_clean2 |>
ggplot2::ggplot(
ggplot2::aes(sample = systolic)
) +
ggplot2::stat_qq(
ggplot2::aes(color = "NHANES participant")
) +
ggplot2::stat_qq_line(
ggplot2::aes(linetype = "Normally distributed"),
linewidth = 1
) +
ggplot2::labs(
x = "Theoretical normal distribution",
y = "Observes systolic blood pressure (mmHG)"
) +
ggplot2::scale_color_manual(
name = "",
values = ("NHANES participant" = "purple3")
) +
ggplot2::scale_linetype_manual(
name = "",
values = ("Normally distributed" = "solid")
)
(This is the replication of book Figure 6.12, that has no accompanying code.)
This q-q-plot shows clearly that we have failed the assumption of a normal distribution. But often the visible check is not so obvious. Then we would need a statistical test to check for normality.
Normality has to be checked for each group for the independent-samples t-test (and dependent-samples t-test).
Example 6.9 : Independent-samples t-test: Checking the normality assumption visually
R Code 6.33 : Checking normality with histogram
bp_clean <- base::readRDS("data/chap06/bp_clean.rds")
bp_clean |>
ggplot2::ggplot(
ggplot2::aes(x = systolic)
) +
ggplot2::geom_histogram(
fill = "purple3",
col = "white",
na.rm = TRUE,
bins = 30
) +
ggplot2::facet_grid(
cols = ggplot2::vars(sex)
) +
ggplot2::labs(
x = "Systolic blood pressure (mmHg)",
y = "NHANES participants"
)
Both histograms look right-skewed.
R Code 6.34 : Checking normality with Q-Q plot
bp_clean |>
tidyr::drop_na(systolic) |>
ggplot2::ggplot(
ggplot2::aes(sample = systolic)
) +
ggplot2::stat_qq(
ggplot2::aes(color = "NHANES participant"),
alpha = 0.5
) +
ggplot2::stat_qq_line(
ggplot2::aes(linetype = "Normally distributed"),
linewidth = 1
) +
ggplot2::facet_grid(
cols = ggplot2::vars(sex)
) +
ggplot2::labs(
x = "Theoretical normal distribution",
y = "Observes systolic blood pressure (mmHG)"
) +
ggplot2::scale_color_manual(
name = "",
values = ("NHANES participant" = "purple3")
) +
ggplot2::scale_linetype_manual(
name = "",
values = ("Normally distributed" = "solid")
)
Harris uses in the book a more complicated code passage to build the line for the normal distribution. Instead using the function ggplot2::stat_qq_line() she calculates intercept and slopes and applies the ggplot2::abline() function.
The q-q plot shows that both blood pressures (males and females) are not normally distributed.
Here we are going to check the distribution of the difference between the first (systolic) and second blood pressure measure (systolic2).
Example 6.10 : Dependent-samples t-test: Checking the normality assumption visually
R Code 6.35 : Checking the normality assumption visually with a histogram and an overlaid normal distribution
The histogram looks like a normal distribution. But let’s verify this rsult with a q-q plot.
R Code 6.36 : Checking the normality assumption visually with a quantile-quantile plot
bp_clean3 |>
ggplot2::ggplot(
ggplot2::aes(sample = diff_syst)
) +
ggplot2::stat_qq(
ggplot2::aes(color = "NHANES participant")
) +
ggplot2::stat_qq_line(
ggplot2::aes(linetype = "Normally distributed"),
linewidth = 1
) +
ggplot2::labs(
x = "Theoretical normal distribution",
y = "Observes systolic blood pressure (mmHG)"
) +
ggplot2::scale_color_manual(
name = "",
values = ("NHANES participant" = "purple3")
) +
ggplot2::scale_linetype_manual(
name = "",
values = ("Normally distributed" = "solid")
)
Bad news! This is a disappointing result: Th Q-Q plot shows that we do not have normal distribution. It just looked like one as the lower and higher values cancel each out.
For all t-tests we applied in this chapter we got the clear result that they all failed the normality assumption.
But often the visual inspection is not conclusive. Therefore it is important also to apply statistical tests.
Different statistical checks for normality for different situations
Different statistical checks of normality are useful in different situations.
Resource 6.4 Packages with tests for skewness
After another research I will — in addition to Section 2.6.2 — propose other packages with tests for skewness. A comparison shows that these tests are much more popular given their download figures as the two packages (Package Profile A.80 and Package Profile A.85) I have already reviewed. All of the mentioned packages have functions for tests for kurtosis as well.
#> # A tibble: 6 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 e1071 10808 2024-03-21 2024-03-27
#> 2 psych 7565 2024-03-21 2024-03-27
#> 3 datawizard 3618 2024-03-21 2024-03-27
#> 4 moments 1792 2024-03-21 2024-03-27
#> 5 semTools 475 2024-03-21 2024-03-27
#> 6 statpsych 12 2024-03-21 2024-03-27Example 6.11 : Skewness of systolic blood pressure
R Code 6.37 : Compute skewness of systolic blood pressure with {semTools}
semTools::skew(bp_clean$systolic)#> Warning in semTools::skew(bp_clean$systolic): Missing observations are removed
#> from a vector.#> skew (g1) se z p
#> 1.07037232 0.02897841 36.93689298 0.00000000Comparing the z-value in Table 2.10 we see that our value is 5 times higher than 7, meaning our distribution is highly right skewed. With a p-value < 0.001 we have to reject the null (that we have a normal distribution).
R Code 6.38 : Compute skewness of systolic blood pressure with {e1071}
e1071::skewness(
x = bp_clean$systolic,
na.rm = TRUE,
type = 2)#> [1] 1.070372There are three slightly different types of skewness computation with {e1071}.
If we compare with the skewness computation of {semTools} we see that SAS/SPSSS calculation is used with type 2 in {e1071}.
We can’t assess the skewness with Table 2.10, because {e1071} does not calculate the z- and p-value. We therefore add another assessment criteria for skewness:
Assessment 6.3 : Magnitude of skewness
A value above 1.0 suggests a strong right skewness.
R Code 6.39 : Compute skewness of systolic blood pressure with {moments}
moments::skewness(
x = bp_clean$systolic,
na.rm = TRUE)#> [1] 1.070148This slightly different value would be the result in {e1071} with type = 1 (older textbooks).
R Code 6.40 : Compute skewness of systolic blood pressure with {datawizard}
#> Skewness | SE
#> ----------------
#> 1.070 | 0.029
#> Skewness | SE | z | p
#> -----------------------------
#> 1.070 | 0.029 | 36.952 | 0There are two versions with {datawizard}. One with just the skewness and standard error, another one — similar as with {semTools} — with skewness, standard error, z-value and p-value.
R Code 6.41 : Compute skewness of systolic blood pressure with {psych}
#> [1] 1.070372
#> Call: psych::mardia(x = bp_clean$systolic)
#>
#> Mardia tests of multivariate skew and kurtosis
#> Use describe(x) the to get univariate tests
#> n.obs = 7145 num.vars = 1
#> b1p = 1.14 skew = 1363.19 with probability <= 2.1e-298
#> small sample skew = 1364.33 with probability <= 1.2e-298
#> b2p = 4.99 kurtosis = 34.33 with probability <= 0There are two different approaches with {psych}:
R Code 6.42 : Statistical test of normality for systolic blood pressure by sex
bp_clean |>
tidyr::drop_na(systolic) |>
dplyr::group_by(sex) |>
dplyr::summarize(
skew = semTools::skew(object = systolic)[1],
se_skew = semTools::skew(object = systolic)[2],
z_skew = semTools::skew(object = systolic)[3],
p_skew = semTools::skew(object = systolic)[4],
e1071 = e1071::skewness(x = systolic, type = 2),
moments = moments::skewness(x = systolic),
psych = psych::skew(x = systolic, type = 2)
)#> # A tibble: 2 × 8
#> sex skew se_skew z_skew p_skew e1071 moments psych
#> <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 Male 1.06 0.0414 25.6 0 1.06 1.06 1.06
#> 2 Female 1.12 0.0406 27.6 0 1.12 1.12 1.12Our data fail the assumption for normality for the independent-samples t-test.
The columns 2-5 are calculated with {semTools}. For comparison I have added other the result of other packages as well.
R Code 6.43 : Statistical test of normality for difference of systolic blood pressure measure 1 and 2
semTools::skew(object = bp_clean3$diff_syst)#> skew (g1) se z p
#> 2.351789e-01 2.906805e-02 8.090632e+00 6.661338e-16The skewness value is much lower but still over the limit of 7 for our sample size. So the null has to be rejected (p-value < 0001).
Besides of testing skewness there are also other statistical tests available. I have researched the following omnibus tests:
stats::shapiro.test(). The test statistic of the Shapiro-Francia test is simply the squared correlation between the ordered sample values and the (approximated) expected ordered quantiles from the standard normal distribution. The Shapiro-Wilk test is known to perform well. But this test is very sensitive and can only be applied when the non-missing values are between 5 and 5000. With samples greater than 5,000 it will always result in statistically significant p-values. Therefore we can’t use it as we have a sample size of about 7000 non-missing values.nortest::lillie.test() is the same as that obtained from stats::ks.test(), it is not correct to use the p-value from the latter for the composite hypothesis of normality (mean and variance unknown), since the distribution of the test statistic is different when the parameters are estimated.Resource 6.5 Packages for the Anderson-Darling test
#> # A tibble: 4 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 nortest 2448 2024-03-21 2024-03-27
#> 2 DescTools 2099 2024-03-21 2024-03-27
#> 3 kSamples 420 2024-03-21 2024-03-27
#> 4 cmstatr 20 2024-03-21 2024-03-27I will use only the two most downloaded packages: {DescTools} because I have already installed it (see Package Profile A.15) and {nortest} because it is a specialized package for test for normality and most articles refer to its usage.
Example 6.12 : Anderson-Darling Test of Normality
R Code 6.44 : One-sample t-test: Test of normality using the Anderson-Darling test with the {nortest} package
nortest::ad.test(bp_clean$systolic)#>
#> Anderson-Darling normality test
#>
#> data: bp_clean$systolic
#> A = 88.39, p-value < 2.2e-16We’ve got a very tiny p-value so we have to reject the null hypothesis that the systolic blood pressure is normally distributed.
R Code 6.45 : One-sample t-test: Test of normality using the Anderson-Darling test with the {DescTools} package
DescTools::AndersonDarlingTest(
x = bp_clean2$systolic,
null = "pnorm",
mean = mean(bp_clean2$systolic),
sd = sd(bp_clean2$systolic)
)#>
#> Anderson-Darling test of goodness-of-fit
#> Null hypothesis: Normal distribution
#> with parameters mean = 120.539, sd = 18.617
#>
#> data: bp_clean2$systolic
#> An = 88.39, p-value = 8.397e-08Again, we have to reject the Null.
The {DescTools} packages has the advantage that it is a general test of goodness-of-fit. Besides normality it could also be used to test other kind of distributions.
R Code 6.46 : Dependent-samples t-test: Test of normality using the Anderson-Darling test with {nortest} and {DescTools} package
glue::glue("Computation with nortest package")
nortest::ad.test(bp_clean$diff_syst)
glue::glue(" ")
glue::glue("###################################################")
glue::glue("Computation with DescTools package")
DescTools::AndersonDarlingTest(
x = bp_clean3$diff_syst,
null = "pnorm",
mean = mean(bp_clean3$diff_syst),
sd = sd(bp_clean3$diff_syst)
)#> Computation with nortest package
#>
#> Anderson-Darling normality test
#>
#> data: bp_clean$diff_syst
#> A = 69.694, p-value < 2.2e-16
#>
#>
#> ###################################################
#> Computation with DescTools package
#>
#> Anderson-Darling test of goodness-of-fit
#> Null hypothesis: Normal distribution
#> with parameters mean = 0.545, sd = 4.898
#>
#> data: bp_clean3$diff_syst
#> An = 69.694, p-value = 8.45e-08Both tests agree, even their p-values differ: We have to reject the Null.
Homogeneity of variances is another assumption valid for the independent t-test. The data should be equally spread out in each group. Although we had used the Welch’s version of the t-test, which does not require homogeneity of variances, we will apply Levene’s test anyway. (An additional visual inspection via group-specific boxplots would helpful to gain a visual understanding of Levene’s test results.)
Resource 6.6 Packages for Levene’s test
{car}: Companion to Applied Regression {DescTools}: Tools for Descriptive Statistics {misty}: Miscellaneous Functions ‘T. Yanagida’ {rstatix}: Pipe-Friendly Framework for Basic Statistical Tests
#> # A tibble: 4 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 car 13044 2024-03-21 2024-03-27
#> 2 rstatix 5551 2024-03-21 2024-03-27
#> 3 DescTools 2099 2024-03-21 2024-03-27
#> 4 misty 804 2024-03-21 2024-03-27The {car} packages is most of time used for Levene’s test. There is also a specific tidy() function in {broom} for the results of car::leveneTest().
Example 6.13 : Levene’s test with different packages
R Code 6.47 : Testing for equal variances for systolic by sex with {car}
car::leveneTest(
y = systolic ~ sex,
data = bp_clean)#> Levene's Test for Homogeneity of Variance (center = median)
#> Df F value Pr(>F)
#> group 1 3.552 0.05952 .
#> 7143
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The variances of systolic blood pressure for men and women are not statistically significantly different (p = .06), and the independent-samples t-test meets the assumption of homogeneity of variances.
R Code 6.48 : Testing for equal variances for systolic by sex with {DescTools}
DescTools::LeveneTest(
formula = systolic ~ sex,
data = bp_clean
)#> Levene's Test for Homogeneity of Variance (center = median)
#> Df F value Pr(>F)
#> group 1 3.552 0.05952 .
#> 7143
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Exactyl the same result as with car::leveneTest().
R Code 6.49 : Testing for equal variances for systolic by sex with {rstatix}
rstatix::levene_test(
formula = systolic ~ sex,
data = bp_clean
)#> # A tibble: 1 × 4
#> df1 df2 statistic p
#> <int> <int> <dbl> <dbl>
#> 1 1 7143 3.55 0.0595Again exact the same result but this time it returns a tibble which is easier to work with.
R Code 6.50 : Testing for equal variances for systolic by sex with {misty}
misty::test.levene(
formula = systolic ~ sex,
data = bp_clean,
plot = TRUE
)
#> Levene's Test based on the Median
#>
#> Null hypothesis H0: sigma2.1 = sigma2.2
#> Alternative hypothesis H1: sigma2.1 != sigma2.2
#>
#> Group n nNA M SD Var Low Upp
#> Male 3498 0 122.18 18.15 329.30 307.52 353.01
#> Female 3647 0 118.97 18.93 358.23 335.99 382.36
#>
#> Df Sum Sq Mean Sq F pval
#> Group 1 558.73 558.73 3.55 0.060
#> Residuals 7143 1123605.77 157.30This is most detailed computation! The function has 35 arguments, most of them are dedicated to the appearance of the plot.
The systolic data fails the assumption of normality for all different kind of t-tests. This means that we cannot rely resp. apply the t-test as we had done in this chapter! Practically we would need to start with our analysis anew.
These are our alternatives:
If the normality assumption of the one-sample t-test fails, the median could be examined rather than the mean — just like for descriptive statistics when the variable is not normally distributed.
The median for the systolic distribution is 118. We will compare this value to our hypothetical population median of 120 mmHG. So our null hypothesis is, that the median of 118 is coming from a population with a median of 120 mmHG systolic blood pressure.
Resource 6.7 Packages with function for the sign test
#> # A tibble: 3 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 rstatix 5551 2024-03-21 2024-03-27
#> 2 DescTools 2099 2024-03-21 2024-03-27
#> 3 BSDA 625 2024-03-21 2024-03-27Generally I prefer packages where I am using several functions and with a high number of downloads. This is the case for {rstatix} and {DescTools}. Additionally I will also test the {BSDA} recommendation of the book.
Example 6.14 : Applying sign test to the systolic blood pressure of NHANES participants
R Code 6.51 : Compare observed median of systolic blood pressure with 120 mmHG using {BSDA}
BSDA::SIGN.test(
x = bp_clean$systolic,
md = 120
)#>
#> One-sample Sign-Test
#>
#> data: bp_clean$systolic
#> s = 3004, p-value < 2.2e-16
#> alternative hypothesis: true median is not equal to 120
#> 95 percent confidence interval:
#> 116 118
#> sample estimates:
#> median of x
#> 118
#>
#> Achieved and Interpolated Confidence Intervals:
#>
#> Conf.Level L.E.pt U.E.pt
#> Lower Achieved CI 0.9477 116 118
#> Interpolated CI 0.9500 116 118
#> Upper Achieved CI 0.9505 116 118We have reason to reject the null hypothesis, e.g. our observed median is not coming from a population median of 120 mmHG. Our sample came likely from a population where the median systolic blood pressure was between 116 and 118 mmHG.
The median systolic blood pressure for NHANES participants was 118 mmHG. A sign test comparing the median to a hypothesized median of 120 mmHG had a statistically significant (s = 3004; p < .05) result. The sample with a median systolic blood pressure of 118 was unlikely to have come from a population with a median systolic blood pressure of 120. The 95% confidence interval indicates this sample likely came from a population where the median systolic blood pressure was between 116 and 118. This suggests that the median systolic blood pressure in the U.S. population is between 116 and 118.
R Code 6.52 : Compare observed median of systolic blood pressure with 120 mmHG using {DescTools}
DescTools::SignTest(
x = bp_clean$systolic,
mu = 120
)#>
#> One-sample Sign-Test
#>
#> data: bp_clean$systolic
#> S = 3004, number of differences = 6844, p-value < 2.2e-16
#> alternative hypothesis: true median is not equal to 120
#> 95 percent confidence interval:
#> 116 118
#> sample estimates:
#> median of the differences
#> 118R Code 6.53 : Compare observed median of systolic blood pressure with 120 mmHG using {rstatix}
rstatix::sign_test(
data = bp_clean,
formula = systolic ~ 1,
mu = 120,
detailed = TRUE
)#> # A tibble: 1 × 12
#> estimate .y. group1 group2 n statistic p df conf.low conf.high
#> * <dbl> <chr> <chr> <chr> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 118 syst… 1 null … 7145 3004 5.21e-24 6844 116 118
#> # ℹ 2 more variables: method <chr>, alternative <chr>As always {rstatix} returns a tibble and has all the details. It is most of the time my favorite package for tests.
The Wilcoxon signed-ranks test is an alternative to the dependent-samples t-test when the continuous variable is not normally distributed. The Wilcoxon test determines if the differences between paired values of two related samples are symmetrical around zero. That is, instead of comparing the mean difference to zero, the test compares the distribution of the differences around zero.
Procedure 6.3 : Wilcoxon signed-ranks test
Some confusing details:
W, although it was sometimes reported as T and called the Wilcoxon T-test.
W.W.W is approximately normal when the sample size is more than 20.W is statistically significant.stats::wilcox.test() function shows neither T nor W bit V in its output. This is the sum of the ranks of positive differences. V would be the same as W when the sum of the ranks of positive differences was highest, but different from W when the sum of the ranks for negative differences was highest. In practical terms this difference is not important as I will take information for the decision (to reject or not to rejet the Null) from the p-value.Write the null and alternate hypotheses:
Compute the test statistic.
Example 6.15 : Compute Wilcoxon test with different packages
R Code 6.54 : Compute Wilcoxon test with stats::wilcox.test()
stats::wilcox.test(
x = bp_clean$systolic,
y = bp_clean$systolic2 ,
conf.int = TRUE,
paired = TRUE)#>
#> Wilcoxon signed rank test with continuity correction
#>
#> data: bp_clean$systolic and bp_clean$systolic2
#> V = 9549959, p-value < 2.2e-16
#> alternative hypothesis: true location shift is not equal to 0
#> 95 percent confidence interval:
#> 6.335319e-05 9.999790e-01
#> sample estimates:
#> (pseudo)median
#> 2.923076e-05R Code 6.55 : Compute Wilcoxon test with rstatix::pairwise_wilcox_test()
wilcox_rstatix <- bp_clean |>
dplyr::select(systolic, systolic2) |>
tidyr::drop_na() |>
tidyr::pivot_longer(
cols = c("systolic", "systolic2"),
names_to = "treatment",
values_to = "value") |>
rstatix::pairwise_wilcox_test(
formula = value ~ treatment,
paired = TRUE,
detailed = TRUE
)
knitr::kable(wilcox_rstatix)systolic) and time 2 (systolic2)
| estimate | .y. | group1 | group2 | n1 | n2 | statistic | p | conf.low | conf.high | method | alternative | p.adj | p.adj.signif |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.92e-05 | value | systolic | systolic2 | 7101 | 7101 | 9549959 | 0 | 6.34e-05 | 0.999979 | Wilcoxon | two.sided | 0 | **** |
Similar as in Example 6.7 we had to apply tidyr::pivot_longer() to get the data into the right shape.
Review and interpret the test statistics: Calculate the probability that your test statistic is at least as big as it is if there is no relationship (i.e., the null is true).
The p-value is well below .05.
Conclude and write report.
We have to reject the NULL and assume the alternative: The distribution of the difference between the systolic blood pressure measures taken at Time 1 and Time 2 in the U.S. population is not symmetric around zero.
We used a Wilcoxon signed-ranks test to determine whether the distribution of the difference in systolic blood pressure measured at Time 1 and Time 2 was symmetrical around zero. The resulting test statistic and p-value indicated that the sample likely came from a population where the differences were not symmetrical around zero (p < .05). That is, we found a significant difference between the first and second blood pressure measures.
WATCH OUT! Interpreting the results as comparing medians can be misleading
are often interpreted as testing for equal medians.
While none of these tests examine medians directly, the ordering and ranking of values is similar to how medians are identified, so there is logic to this interpretation. However, if the distribution shape or spread (or both) are different, interpreting the results as comparing medians can be misleading.
Conduct visual and descriptive analyses before (or with) these tests to make sure you interpret the results accurately.
Mann-Whitney U test is used when the t-test assumption of normality for independent group is not met. It is considered to be the non-parametric equivalent to the two-sample independent t-test. The U test also relaxes the variable type assumption and can be used for ordinal variables in addition to continuous variables. It works similar as the Wilcoxon signed-ranks test:
Write the null and alternate hypotheses:
Compute the test statistic.
Example 6.16 : Compute Mann-Whitney U test with different packages
R Code 6.56 : Compute Mann-Whitney U test with stats::wilcox.test()
mann_whitney_stats <- stats::wilcox.test(
formula = bp_clean$systolic ~ bp_clean$sex,
conf.int = TRUE,
paired = FALSE)
mann_whitney_stats#>
#> Wilcoxon rank sum test with continuity correction
#>
#> data: bp_clean$systolic by bp_clean$sex
#> W = 7186882, p-value < 2.2e-16
#> alternative hypothesis: true location shift is not equal to 0
#> 95 percent confidence interval:
#> 2.000038 4.000012
#> sample estimates:
#> difference in location
#> 3.99998WATCH OUT! Mann-Whitney U test is also called Wilcoxon rank sum test! And this is not the same as Wilcoxon signed-ranks test.
For more information on the different Wilcoxon tests read Wilcoxon Test in R (Kassambara, n.d.).
R Code 6.57 : Compute Mann-Whitney U test with rstatix::pairwise_wilcox_test()
wilcox_rstatix <- bp_clean |>
dplyr::select(systolic, sex) |>
tidyr::drop_na() |>
rstatix::pairwise_wilcox_test(
formula = systolic ~ sex,
paired = FALSE,
detailed = TRUE
)
knitr::kable(wilcox_rstatix)| estimate | .y. | group1 | group2 | n1 | n2 | statistic | p | conf.low | conf.high | method | alternative | p.adj | p.adj.signif |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.99998 | systolic | Male | Female | 3498 | 3647 | 7186882 | 0 | 2.000038 | 4.000012 | Wilcoxon | two.sided | 0 | **** |
\[ r = \frac{z}{\sqrt{n}} \tag{6.7}\]
The Pearson correlation coefficient r is calculated as z-score divided by square root of the total observations.
Assessment 6.4 : Effect size r of the Mann-Whitney U test
:::::{.my-example} :::{.my-example-header} :::::: {#exm-chap06-effect-size-mann-whitney-u-test} : Compute effect size for the Mann-Whitney U test :::::: ::: ::::{.my-example-container}
R Code 6.58 : Compute effect size for the Mann-Whitney U test manually
:::::
Review and interpret the test statistics: Calculate the probability that your test statistic is at least as big as it is if there is no relationship (i.e., the null is true).
As the p-value < 2.2e-16 with a small effect size of r = 0.11 is statistically significant, we reject the Null.
Conclude and write report.
A Mann-Whitney U test comparing systolic blood pressure for males and females in the United States found a statistically significant difference between the two groups (p < .05). Histograms demonstrated the differences, with notably more females with systolic blood pressure below 100 compared to males along with some other differences. The effect size was small, r = .11, indicating a weak but statistically significant relationship between sex and systolic blood pressure.
All the tests under the heading “Alternate tests” (Section 6.10) discussed so far are alternatives for a failed normality assumption. Kolmogorov-Smirnov test is an alternative when the assumption of equal variances (homogeneity of variances) has failed.
When the variances are unequal, the homogeneity of variances assumption is not met, whether or not the normality assumption is met. The larger variance has a bigger influence on the size of the t-statistic, so one group is dominating the t-statistic calculations.
Although we have used the Welch’s t-test the Kolmogorov-Smirnov test is an alternative test when both (the normality and the variance assumption) have failed.
Write the null and alternate hypotheses:
Compute the test statistic.
R Code 6.60 : Compute the Kolmogorov-Smirnov test with {stats}
#> Warning in ks.test.default(x = male_sys, y = female_sys): p-value will be
#> approximate in the presence of ties#>
#> Asymptotic two-sample Kolmogorov-Smirnov test
#>
#> data: male_sys and female_sys
#> D = 0.11408, p-value < 2.2e-16
#> alternative hypothesis: two-sidedMost of the examples in the Internet uses the base R version with stats::ks.test() for the Kolmogorov-Smirnov test. I have found with {FSA} and {dgof} two other packages with this test, but as they do not differ with their input parameters nor with their results I will stick with the {stats} function.
Review and interpret the test statistics: Calculate the probability that your test statistic is at least as big as it is if there is no relationship (i.e., the null is true).
The p-value is well below .05.
Conclude and write report.
The test statistic, D, is the maximum distance between the two empirical cumulative distribution functions (ECDFs). ECDFs are a special type of probability distribution showing the cumulative probability of the values of a variable. We can examine the difference between the ECDFs for systolic blood pressure of males and females in the sample with the following graph:
R Code 6.61 : Compute empirical cumulative distributions (ECDFs)
bp_clean |>
ggplot2::ggplot(
ggplot2::aes(
x = systolic,
color = sex
)
) +
ggplot2::stat_ecdf(
geom = "step",
linewidth = 1,
na.rm = TRUE
) +
ggplot2::labs(
x = "Systolic blood pressure (mmHg)",
y = "Cumulative probability"
) +
ggplot2::scale_color_manual(
values = c(
"Male" = "gray",
"Female" = "purple3"
)
)
At the widest gap between these two curves, males and females were \(.11\) apart, giving a test statistic of \(D = .11\).
A K-S test comparing systolic blood pressure for males and females found a statistically significant difference between the two groups (D = .11; p < .05). This sample likely came from a population where the distribution of systolic blood pressure was different for males and females.
Resource 6.8 Miscellaneous packages for (possible) alternative tests
During my research for packages with functions for alternative test if the t-test assumption fail I came about several very interesting packages. Some of them work together with {ggplot2}, so that these statistics can be plotted easily as well.
#> # A tibble: 4 × 4
#> package average from to
#> <chr> <dbl> <date> <date>
#> 1 ggstats 3191 2024-03-21 2024-03-27
#> 2 rcompanion 768 2024-03-21 2024-03-27
#> 3 statsExpressions 547 2024-03-21 2024-03-27
#> 4 ggstatsplot 380 2024-03-21 2024-03-27| term | definition |
|---|---|
| Anderson-Darling | The Anderson-Darling Goodness of Fit Test (AD-Test) is a measure of how well your data fits a specified distribution. It’s commonly used as a test for normality. (<a href="https://www.statisticshowto.com/anderson-darling-test/">Statistics How-To</a>) |
| Boxplots | Boxplots are a visual representation of data that shows central tendency (usually the median) and spread (usually the interquartile range) of a numeric variable for one or more groups; boxplots are often used to compare the distribution of a continuous variable across several groups. (SwR, Glossary) |
| Cohen’s d | Cohen’s d is a standardized effect size for measuring the difference between two group means. It is frequently used to compare a treatment to a control group. It can be a suitable effect size to include with t-test and ANOVA results. (<a href= "https://statisticsbyjim.com/basics/cohens-d/">Statistics by Jim</a>) |
| Degrees of Freedom | Degree of Freedom (df) is the number of pieces of information that are allowed to vary in computing a statistic before the remaining pieces of information are known; degrees of freedom are often used as parameters for distributions (e.g., chi-squared, F). (SwR, Glossary) |
| ECDF | In statistics, an empirical distribution function (commonly also called an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. (<a href="https://en.wikipedia.org/wiki/Empirical_distribution_function">Wikipedia</a>) A CDF is a hypothetical model of a distribution, the ECDF models empirical (i.e. observed) data. (<a href="https://www.statisticshowto.com/empirical-distribution-function/">Statistics How To</a>) |
| Effect Size | Effect size is a measure of the strength of a relationship; effect sizes are important in inferential statistics in order to determine and communicate whether a statistically significant result has practical importance. (SwR, Glossary) |
| Homogeneity of Variances | Homogeneity of variances is equal variances among groups; homogeneity of variance is one of the assumptions tested for independent and dependent t-tests and analysis of variance. (SwR, Glossary) |
| Independent | Independent-samples t-test or unpaired sample t-test is an inferential test comparing two independent means. (SwR, Glossary) |
| Kolmogorov-Smirnov | The Kolmogorov-Smirnov test is used when the assumption of equal variances (homogeneity of variances) fails for the independent-samples t-test; the test compares the distributions of the groups rather than their means. (SwR, Glossary) |
| Kurtosis | Kurtosis is a measure of how many observations are in the tails of a distribution; distributions that look bell-shaped, but have a lot of observations in the tails (platykurtic) or very few observations in the tails (leptokurtic) (SwR, Glossary) |
| Leptokurtic | Leptokurtic is a distribution of a numeric variable that has many values clustered around the middle of the distribution; leptokurtic distributions often appear tall and pointy compared to mesokurtic or platykurtic distributions. (SwR, Glossary) |
| Mann-Whitney | Mann-Whitney U test, also called Wilcoxon rank sum test, is an alternative for comparing a numeric or ordinal variable across two groups when the independent-samples t-test assumption of normality is not met. (SwR, Glossary) |
| Omnibus | An omnibus is a statistical test that identifies that there is some relationship going on between variables, but not what that relationship is. (SwR, Glossary) |
| One-sample | One-sample t-test, also known as the single-parameter t-test or single-sample t-test, is an inferential statistical test comparing the mean of a numeric variable to a population or hypothesized mean. (SwR, Glossary) |
| Outcome | Outcome is the variable being explained or predicted by a model; in linear and logistic regression, the outcome variable is on the left-hand side of the equal sign. (SwR, Glossary) |
| p-value | The p-value is the probability that the test statistic is at least as big as it is under the null hypothesis (SwR, Glossary) |
| Paired | Dependent-samples test or paired-samples t-test is an inferential test comparing two related means . (SwR, Glossary) |
| Platykurtic | Platykurtic is a distribution of a numeric variable that has more observations in the tails than a normal distribution would have; platykurtic distributions often look flatter than a normal distribution. (SwR, Glossary) |
| Pooled Variance | Pooled variance is the assumption that the variances in two groups are equal, so these variances are combined ('pooled') (SwR, Glossary). |
| Predictor Variable | Predictor variable -- also known sometimes as the independent or explanatory variable -- is the counterpart to the response or dependent variable. Predictor variables are used to make predictions for dependent variables. ([DeepAI](https://deepai.org/machine-learning-glossary-and-terms/predictor-variable), [MiniTab](https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistical-modeling/regression/supporting-topics/basics/what-are-response-and-predictor-variables/)) |
| Q-Q-Plot | A quantile-quantile plot is a visualization of data using probabilities to show how closely a variable follows a normal distribution. (SwR, Glossary) This plot is made up of points below which a certain percentage of the observations fall. On the x-axis are normally distributed values with a mean of 0 and a standard deviation of 1. On the y-axis are the observations from the data. If the data are normally distributed, the values will form a diagonal line through the graph. (SwR, chapter 6) |
| Quantile | Quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities (<a href="https://en.wikipedia.org/wiki/Quantile">Wikipedia</a>) |
| Rejection Region | Rejection region is the area under the curve of a sampling distribution where the probability of obtaining a value is very small, often below 5%; the rejection region is in the end of the tail or tails of the distribution. (SwR, Glossary) |
| Shapiro-Wilk | The Shapiro-Wilk test is a statistical test to determine or confirm whether a variable has a normal distribution; it is sensitive to small deviations from normality and not useful for sample sizes above 5,000 because it will nearly always find non-normality. (SwR, Glossary) |
| Sign-Test | Sign-test is a a statistical test that compares the median of a variable to a hypothesized or population value; used in lieu of the one-sample t-test when the t-test assumptions are not met. (SwR, Glossary) |
| Skewness | Skewness is the extent to which a variable has extreme values in one of the two tails of its distribution (SwR, Glossary) |
| Standard Deviation | The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is the square root of its variance. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter $\sigma$ (sigma), for the population standard deviation, or the Latin letter $s$ for the sample standard deviation. ([Wikipedia] |
| Standard Error | The standard error (SE) of a statistic is the standard deviation of its [sampling distribution]. If the statistic is the sample mean, it is called the standard error of the mean (SEM). (<a href="https://en.wikipedia.org/wiki/Standard_error">Wikipedia</a>) The standard error is a measure of variability that estimates how much variability there is in a population based on the variability in the sample and the size of the sample. (SwR, Glossary) |
| T-Statistic | The T-Statistic is used in a T test when you are deciding if you should support or reject the null hypothesis. It’s very similar to a Z-score and you use it in the same way: find a cut off point, find your t score, and compare the two. You use the t statistic when you have a small sample size, or if you don’t know the population standard deviation. (<a href="https://www.statisticshowto.com/t-statistic/">Statistics How-To</a>) |
| Welch-T | Welch’s t-test is a variation on the Student’s t-test that does not assume equal variances in group (SwR, Glossary). |
| Wilcoxon | Wilcoxon signed-ranks test is an alternative to the dependent-samples t-test when the continuous variable is not normally distributed; it uses ranks to determine whether the values of a numeric variable are different across two related groups. (SwR, Glossary) |
| Z-score | A z-score (also called a standard score) gives you an idea of how far from the mean a data point is. But more technically it’s a measure of how many standard deviations below or above the population mean a raw score is. (<a href="https://www.statisticshowto.com/probability-and-statistics/z-score/#Whatisazscore">StatisticsHowTo</a>) |
Session Info
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