Category Archives: Applied Statistics

Supplier Quality Management: Seeking Test Data

Do you have, or have you had, a supplier selection problem to solve? I have some algorithms I’ve been working on to help you make better decisions about what suppliers to choose — and how to monitor performance over time. I’d like to test and refine them on real data. If anyone has data that you’ve used to select suppliers in the past 10 years, or have data that you’re working with right now to select suppliers, or have a colleague who may be able to share this data — that’s what I’m interested in sourcing.

Because this data can sometimes be proprietary and confidential, feel free to blind the names or identifying information for the suppliers — or I can do this myself (no suppliers, products, or parts will be named when I publish the results). I just need to be able to tell them apart. Tags like Supplier A or Part1SupplierA are fine. I’d prefer if you blinded the data, but I can also write scripts to do this and have you check them before I move forward.

Desired data format is CSV or Excel. Text files are also OK, as long as they clearly identify the criteria that you used for supplier selection. Email me at myfirstname dot mylastname at gmail if you can help out — and maybe I can help you out too! Thanks.

 

Where Do Z-Score Tables Come From? (+ how to make them in R)

z-score-table-iconEvery student learns how to look up areas under the normal curve using Z-Score tables in their first statistics class. But what is less commonly covered, especially in courses where calculus is not a prerequisite, is where those Z-Score tables come from: by evaluating the integral of the equation for the bell-shaped normal curve, usually from -Inf to the z-score of interest. This is the same thing that the R command pnorm does when you provide it with a z-score. Here is the slide presentation I put together to explain the use and origin of the Z-Score table, and how it relates to pnorm and qnorm (the command that lets you input an area to find the z-score at which the area to the left is swiped out). It’s free to use under Creative Commons, and is part of the course materials that is available for use with this 2015 book.

One of the fun things I did was to make my own z-score table in R. I don’t know why anyone would WANT to do this — they are easy to find in books, and online, and if you know how to use pnorm and qnorm, you don’t need one at all. But, you can, and here’s how.

First, let’s create a z-score table just with left-tail areas. Using symmetry, we can also use this to get any areas in the right tail, because the area to the left of any -z is the same as any area to the right of any +z. Even though the z-score table contains areas in its cells, our first step is to create a table just of the z-scores that correspond to each cell:

c0 <- seq(-3.4,0,.1)
c1 <- seq(-3.41,0,.1)
c2 <- seq(-3.42,0,.1)
c3 <- seq(-3.43,0,.1)
c4 <- seq(-3.44,0,.1)
c5 <- seq(-3.45,0,.1)
c6 <- seq(-3.46,0,.1)
c7 <- seq(-3.47,0,.1)
c8 <- seq(-3.48,0,.1)
c9 <- seq(-3.49,0,.1)
z <- cbind(c0,c1,c2,c3,c4,c5,c6,c7,c8,c9)
z

 c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
 [1,] -3.4 -3.41 -3.42 -3.43 -3.44 -3.45 -3.46 -3.47 -3.48 -3.49
 [2,] -3.3 -3.31 -3.32 -3.33 -3.34 -3.35 -3.36 -3.37 -3.38 -3.39
 [3,] -3.2 -3.21 -3.22 -3.23 -3.24 -3.25 -3.26 -3.27 -3.28 -3.29
 [4,] -3.1 -3.11 -3.12 -3.13 -3.14 -3.15 -3.16 -3.17 -3.18 -3.19
 [5,] -3.0 -3.01 -3.02 -3.03 -3.04 -3.05 -3.06 -3.07 -3.08 -3.09
 [6,] -2.9 -2.91 -2.92 -2.93 -2.94 -2.95 -2.96 -2.97 -2.98 -2.99
 [7,] -2.8 -2.81 -2.82 -2.83 -2.84 -2.85 -2.86 -2.87 -2.88 -2.89
 [8,] -2.7 -2.71 -2.72 -2.73 -2.74 -2.75 -2.76 -2.77 -2.78 -2.79
 [9,] -2.6 -2.61 -2.62 -2.63 -2.64 -2.65 -2.66 -2.67 -2.68 -2.69
[10,] -2.5 -2.51 -2.52 -2.53 -2.54 -2.55 -2.56 -2.57 -2.58 -2.59
[11,] -2.4 -2.41 -2.42 -2.43 -2.44 -2.45 -2.46 -2.47 -2.48 -2.49
[12,] -2.3 -2.31 -2.32 -2.33 -2.34 -2.35 -2.36 -2.37 -2.38 -2.39
[13,] -2.2 -2.21 -2.22 -2.23 -2.24 -2.25 -2.26 -2.27 -2.28 -2.29
[14,] -2.1 -2.11 -2.12 -2.13 -2.14 -2.15 -2.16 -2.17 -2.18 -2.19
[15,] -2.0 -2.01 -2.02 -2.03 -2.04 -2.05 -2.06 -2.07 -2.08 -2.09
[16,] -1.9 -1.91 -1.92 -1.93 -1.94 -1.95 -1.96 -1.97 -1.98 -1.99
[17,] -1.8 -1.81 -1.82 -1.83 -1.84 -1.85 -1.86 -1.87 -1.88 -1.89
[18,] -1.7 -1.71 -1.72 -1.73 -1.74 -1.75 -1.76 -1.77 -1.78 -1.79
[19,] -1.6 -1.61 -1.62 -1.63 -1.64 -1.65 -1.66 -1.67 -1.68 -1.69
[20,] -1.5 -1.51 -1.52 -1.53 -1.54 -1.55 -1.56 -1.57 -1.58 -1.59
[21,] -1.4 -1.41 -1.42 -1.43 -1.44 -1.45 -1.46 -1.47 -1.48 -1.49
[22,] -1.3 -1.31 -1.32 -1.33 -1.34 -1.35 -1.36 -1.37 -1.38 -1.39
[23,] -1.2 -1.21 -1.22 -1.23 -1.24 -1.25 -1.26 -1.27 -1.28 -1.29
[24,] -1.1 -1.11 -1.12 -1.13 -1.14 -1.15 -1.16 -1.17 -1.18 -1.19
[25,] -1.0 -1.01 -1.02 -1.03 -1.04 -1.05 -1.06 -1.07 -1.08 -1.09
[26,] -0.9 -0.91 -0.92 -0.93 -0.94 -0.95 -0.96 -0.97 -0.98 -0.99
[27,] -0.8 -0.81 -0.82 -0.83 -0.84 -0.85 -0.86 -0.87 -0.88 -0.89
[28,] -0.7 -0.71 -0.72 -0.73 -0.74 -0.75 -0.76 -0.77 -0.78 -0.79
[29,] -0.6 -0.61 -0.62 -0.63 -0.64 -0.65 -0.66 -0.67 -0.68 -0.69
[30,] -0.5 -0.51 -0.52 -0.53 -0.54 -0.55 -0.56 -0.57 -0.58 -0.59
[31,] -0.4 -0.41 -0.42 -0.43 -0.44 -0.45 -0.46 -0.47 -0.48 -0.49
[32,] -0.3 -0.31 -0.32 -0.33 -0.34 -0.35 -0.36 -0.37 -0.38 -0.39
[33,] -0.2 -0.21 -0.22 -0.23 -0.24 -0.25 -0.26 -0.27 -0.28 -0.29
[34,] -0.1 -0.11 -0.12 -0.13 -0.14 -0.15 -0.16 -0.17 -0.18 -0.19
[35,] 0.0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09

Now that we have slots for all the z-scores, we can use pnorm to transform all those values into the areas that are swiped out to the left of that z-score. This part is easy, and only takes one line. The remaining three lines format and display the z-score table:

zscore.df <- round(pnorm(z),4)
row.names(zscore.df) <- sprintf("%.2f", c0)
colnames(zscore.df) <- seq(0,0.09,0.01)
zscore.df

 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
-3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

You can also draw a picture to go along with your z-score table, so that people remember which area they are looking up:

x <- seq(-4,4,0.1)
y <- dnorm(x)
plot(x,dnorm(x),type="l", col="black", lwd=3)
abline(v=-1,lwd=3,col="blue")
abline(h=0,lwd=3,col="black")
polygon(c(x[1:31],rev(x[1:31])), c(rep(0,31),rev(y[1:31])), col="lightblue")

It looks like this:

z-score-table-icon-small

In the slides, code to produce a giant-tail z-score table is also provided (where the areas are > 50%).

Taking a Subset of a Data Frame in R

I just wrote a new chapter for my students describing how to subset a data frame in R. The full text is available at https://docs.google.com/document/d/1K5U11-IKRkxNmitu_lS71Z6uLTQW_fp6QNbOMMwA5J8/edit?usp=sharing but here’s a preview:

Let’s load in ChickWeight, one of R’s built in datasets. This contains the weights of little chickens at 12 different times throughout their lives. The chickens are on different diets, numbered 1, 2, 3, and 4. Using the str command, we find that there are 578 observations in this data frame, and two different categorical variables: Chick and Diet.

> data(ChickWeight)
> head(ChickWeight)
  weight Time Chick Diet
1     42    0     1    1
2     51    2     1    1
3     59    4     1    1
4     64    6     1    1
5     76    8     1    1
6     93   10     1    1
> str(ChickWeight)
Classes ‘nfnGroupedData’, ‘nfGroupedData’, ‘groupedData’ and 'data.frame':      578 obs. of  4 variables:
 $ weight: num  42 51 59 64 76 93 106 125 149 171 ...
 $ Time  : num  0 2 4 6 8 10 12 14 16 18 ...
 $ Chick : Ord.factor w/ 50 levels "18"<"16"<"15"<..: 15 15 15 15 15 15 15 15 15 15 ...
 $ Diet  : Factor w/ 4 levels "1","2","3","4": 1 1 1 1 1 1 1 1 1 1 ...
 - attr(*, "formula")=Class 'formula' length 3 weight ~ Time | Chick
  .. ..- attr(*, ".Environment")=<environment: R_EmptyEnv> 
 - attr(*, "outer")=Class 'formula' length 2 ~Diet
  .. ..- attr(*, ".Environment")=<environment: R_EmptyEnv> 
 - attr(*, "labels")=List of 2
  ..$ x: chr "Time"
  ..$ y: chr "Body weight"
 - attr(*, "units")=List of 2
  ..$ x: chr "(days)"
  ..$ y: chr "(gm)"

Get One Column: Now that we have a data frame named ChickWeight loaded into R, we can take subsets of these 578 observations. First, let’s assume we just want to pull out the column of weights. There are two ways we can do this: specifying the column by name, or specifying the column by its order of appearance. The general form for pulling information from data frames is data.frame[rows,columns] so you can get the first column in either of these two ways:

ChickWeight[,1]   		# get all rows, but only the first column
ChickWeight[,c("weight")]	# get all rows, and only the column named “weight”

Get Multiple Columns: If you want more than one column, you can specify the column numbers or the names of the variables that you want to extract. If you want to get the weight and diet columns, you would do this:

ChickWeight[,c(1,4)]   		# get all rows, but only 1st and 4th columns
ChickWeight[,c("weight","Diet")]	# get all rows, only “weight” & “Diet” columns

If you want more than one column and those columns are next to each other, you can do this:

ChickWeight[,c(1:3)]

Get One Row: You can get the first row similarly to how you got the first column, and any other row the same way:

ChickWeight[1,]   		# get first row, and all columns
ChickWeight[82,]   		# get 82nd row, and all columns

Get Multiple Rows: If you want more than one row, you can specify the row numbers you want like this:

> ChickWeight[c(1:6,15,18,27),] 
   weight Time Chick Diet      
1      42    0     1    1   
2      51    2     1    1 
3      59    4     1    1    
4      64    6     1    1    
5      76    8     1    1 
6      93   10     1    1    
15     58    4     2    1    
18    103   10     2    1 
27     55    4     3    1    

Using xda with googlesheets in R

Image Credit: Doug Buckley of http://hyperactive.to

Image Credit: Doug Buckley of http://hyperactive.to

Want to do a quick, exploratory data analysis in R of your data that’s stored in a spreadsheet on Google Drive? You’re in luck, because now you can use the new xda package in conjunction with Jenny Bryan‘s googlesheets. There are some quirks, though, and that’s what this post is all about.

Before proceeding, you should review this recent article from R-Bloggers called “Introducing xda”.

First, be sure to install the googlesheets and xda packages. Although googlesheets is on CRAN, xda is not, and you’ll have to bring it in directly from github. You can actually do the same for googlesheets if you like:

install.packages("devtools")
library(devtools)
install_github("jennybc/googlesheets")
install_github("ujjwalkarn/xda")
library(googlesheets)
library(xda)

Next, you’ll have to show R how to access your Google spreadsheet. While you are looking at your spreadsheet, go to File -> Publish to the Web. The URL that’s in the text box is the one you want to capture. Just to make sure it works, copy and paste it into a new browser address window and see if you can display your spreadsheet in your browser.

If you want to import the data at https://docs.google.com/spreadsheets/d/1DO0ksD8d-rn_j2Yn7DQKZDPBrhrvZTpgszewxokjWKU/pubhtml into R, for example, you’ll need to know the spreadsheet’s key. That’s the long string of unintelligible numbers and letters between the “d” and the “pubhtml”. So, my key would be “1DO0ksD8d-rn_j2Yn7DQKZDPBrhrvZTpgszewxokjWKU” — which you’ll see in this next block of code:

> my.gs <- gs_key("1DO0ksD8d-rn_j2Yn7DQKZDPBrhrvZTpgszewxokjWKU") 
> my.data <- gs_read(my.gs) # Retrieves data from googlesheets and places it into an R object. 
> my.df <- as.data.frame(my.data)  # Important! xda needs you to extract only the data in a data frame.

Now, you can access your data. Try head(my.df) to make sure you’ve imported it properly.

Next, it’s time for exploratory data analysis. There are three commands available:

  • numSummary – takes a data frame as an argument, provides descriptive statistics, quantiles, and missing data info for quantitative variables
  • charSummary – takes a data frame as an argument, provides counts, missing data info, and number of unique factors for quantitative variables
  • bivariate – takes a data frame and two quantitative variables as an argument, and performs a quick bivariate analysis (giving this categorical variables, or giving this one categorical and one quantitative variable, will throw an error)

Here’s what happens when you run those commands on the data you just loaded in from your Google spreadsheet:

> numSummary(my.df)
               n   mean    sd median   max  min  mode miss miss%    1%   5%   25%   50%   75%   90%   95%   99%
obs          200 100.50 57.88  100.5 200.0  1.0   1.0    0     0  2.99 11.0  50.8 100.5 150.2 180.1 190.0 198.0
heartrates   200  73.01  7.43   73.0  96.3 53.4  71.2    0     0 56.49 61.2  68.4  73.0  77.4  82.6  85.7  90.3
systolics    200 139.27 29.27  138.0 221.0 59.0 139.0    0     0 79.98 96.0 117.0 138.0 160.0 177.2 188.1 205.0
diastolics   200  87.76  9.74   87.7 116.4 62.4  85.2    0     0 66.01 72.2  81.9  87.7  93.7 100.3 104.5 108.3
bmis         200  25.53  3.06   25.0  33.1 18.4  24.7    0     0 19.00 21.0  23.5  25.0  27.7  29.6  31.2  32.8
ages         200  44.41 14.59   45.0  70.0 18.0  30.0    0     0 18.00 22.0  32.0  45.0  57.0  64.1  67.0  70.0
heartpm      200  72.26  3.55   72.2  83.8 64.2  74.2    0     0 64.72 66.2  69.8  72.2  74.2  76.4  78.7  81.4
fitnesslevel 200   2.62  1.17    3.0   4.0  1.0   4.0    0     0  1.00  1.0   2.0   3.0   4.0   4.0   4.0   4.0

> charSummary(my.df)
          n miss miss% unique
genders 200    0     0      2
smokers 200    0     0      2
group   200    0     0      4

> bivariate(my.df,'heartrates','bmis')
     bin_bmis min_heartrates max_heartrates mean_heartrates
1   (18.3,22]          53.40          85.60           72.80
2   (22,25.7]          55.70          90.70           72.87
3 (25.7,29.4]          60.30          96.30           73.45
4 (29.4,33.1]          56.50          90.30           72.46

Observations:

  • There is a fourth “Plot” command but I couldn’t get it to work on any googlesheetsdata. The xda package is looking for class(range) to be anything other than “function”, which it was for every sheet I attempted to load.
  • There really should be an extra column in xda that displays the enumeration of all the unique values for the factors. It felt great to know how many unique values there were, but I would love to be reminded of what they are too, unless there are too many of them.

Please share your experiences using xda & googlesheets together in the comments! Thanks!

Analytic Hierarchy Process (AHP) using preferenceFunction in ahp

Yesterday, I wrote about how to use gluc‘s new ahp package on a simple Tom-Dick-Harry one level decision making problem using Analytic Hierarchy Process (AHP). One of the cool things about that package is that in addition to specifying the pairwise comparisons directly using Saaty’s scale (below, from https://kristalaace2014.wordpress.com/2014/05/14/w12_al_vendor-evaluation/)…

saaty-scale

…you can also describe each of the Alternatives in terms of descriptive variables which you can use inside a function to make the pairwise comparisons automatically. This is VERY helpful if you have lots of criteria, subcriteria, or alternatives to evaluate!! For example, I used preferenceFunction to compare 55 alternatives using 6 criteria and 4 subcriteria, and was very easily able to create functions to represent my judgments. This was much easier than manually entering all the comparisons.

This post shows HOW I replaced some of my manual comparisons with automated comparisons using preferenceFunction. (The full YAML file is included at the bottom of this post for you to use if you want to run this example yourself.) First, recall that the YAML file starts with specifying the alternatives that you are trying to choose from (at the bottom level of the decision hierarchy) and some variables that characterize those alternatives. I used the descriptions in the problem statement to come up with some assessments between 1=not great and 10=great:

#########################
# Alternatives Section
# THIS IS FOR The Tom, Dick, & Harry problem at
# https://en.wikipedia.org/wiki/Analytic_hierarchy_process_%E2%80%93_leader_example
#
Alternatives: &alternatives
# 1= not well; 10 = best possible
# Your assessment based on the paragraph descriptions may be different.
  Tom:
    age: 50
    experience: 7
    education: 4
    leadership: 10
  Dick:
    age: 60
    experience: 10
    education: 6
    leadership: 6
  Harry:
    age: 30
    experience: 5
    education: 8
    leadership: 6
#
# End of Alternatives Section
#####################################

Here is a snippet from my original YAML file specifying my AHP problem manually ():

  children: 
    Experience:
      preferences:
        - [Tom, Dick, 1/4]
        - [Tom, Harry, 4]
        - [Dick, Harry, 9]
      children: *alternatives
    Education:
      preferences:
        - [Tom, Dick, 3]
        - [Tom, Harry, 1/5]
        - [Dick, Harry, 1/7]
      children: *alternatives

And here is what I changed that snippet to, so that it would do my pairwise comparisons automatically. The functions are written in standard R (fortunately), and each function has access to a1 and a2 (the two alternatives). Recursion is supported which makes this capability particularly useful. I tried to write a function using two of the characteristics in the decision (a1$age and a1$experience) but this didn’t seen to work. I’m not sure whether the package supports it or not. Here are my comparisons rewritten as functions:

  children: 
    Experience:
          preferenceFunction: >
            ExperiencePreference <- function(a1, a2) {
              if (a1$experience < a2$experience) return (1/ExperiencePreference(a2, a1))
              ratio <- a1$experience / a2$experience
              if (ratio < 1.05) return (1)
              if (ratio < 1.2) return (2)
              if (ratio < 1.5) return (3)
              if (ratio < 1.8) return (4)
              if (ratio < 2.1) return (5) return (6) } children: *alternatives Education: preferenceFunction: >
            EducPreference <- function(a1, a2) {
              if (a1$education < a2$education) return (1/EducPreference(a2, a1))
              ratio <- a1$education / a2$education
              if (ratio < 1.05) return (1)
              if (ratio < 1.15) return (2)
              if (ratio < 1.25) return (3)
              if (ratio < 1.35) return (4)
              if (ratio < 1.55) return (5)
              return (5)
            }
          children: *alternatives

To run the AHP with functions in R, I used this code (I am including the part that gets the ahp package, in case you have not done that yet). BE CAREFUL and make sure, like in FORTRAN, that you line things up so that the words START in the appropriate columns. For example, the “p” in preferenceFunction MUST be immediately below the 7th character of your criterion’s variable name.

devtools::install_github("gluc/ahp", build_vignettes = TRUE)
install.packages("data.tree")

library(ahp)
library(data.tree)

setwd("C:/AHP/artifacts")
nofxnAhp <- LoadFile("tomdickharry.txt")
Calculate(nofxnAhp)
fxnAhp <- LoadFile("tomdickharry-fxns.txt")
Calculate(fxnAhp)

print(nofxnAhp, "weight")
print(fxnAhp, "weight")

You can see that the weights are approximately the same, indicating that I did a good job at developing functions that represent the reality of how I used the variables attached to the Alternatives to make my pairwise comparisons. The results show that Dick is now the best choice, although there is some inconsistency in our judgments for Experience that we should examine further. (I have not examined this case to see whether rank reversal could be happening).

> print(nofxnAhp, "weight")
                         levelName     weight
1  Choose the Most Suitable Leader 1.00000000
2   ¦--Experience                  0.54756924
3   ¦   ¦--Tom                     0.21716561
4   ¦   ¦--Dick                    0.71706504
5   ¦   °--Harry                   0.06576935
6   ¦--Education                   0.12655528
7   ¦   ¦--Tom                     0.18839410
8   ¦   ¦--Dick                    0.08096123
9   ¦   °--Harry                   0.73064467
10  ¦--Charisma                    0.26994992
11  ¦   ¦--Tom                     0.74286662
12  ¦   ¦--Dick                    0.19388163
13  ¦   °--Harry                   0.06325174
14  °--Age                         0.05592555
15      ¦--Tom                     0.26543334
16      ¦--Dick                    0.67162545
17      °--Harry                   0.06294121

> print(fxnAhp, "weight")
                         levelName     weight
1  Choose the Most Suitable Leader 1.00000000
2   ¦--Experience                  0.54756924
3   ¦   ¦--Tom                     0.25828499
4   ¦   ¦--Dick                    0.63698557
5   ¦   °--Harry                   0.10472943
6   ¦--Education                   0.12655528
7   ¦   ¦--Tom                     0.08273483
8   ¦   ¦--Dick                    0.26059839
9   ¦   °--Harry                   0.65666678
10  ¦--Charisma                    0.26994992
11  ¦   ¦--Tom                     0.74286662
12  ¦   ¦--Dick                    0.19388163
13  ¦   °--Harry                   0.06325174
14  °--Age                         0.05592555
15      ¦--Tom                     0.26543334
16      ¦--Dick                    0.67162545
17      °--Harry                   0.06294121

> ShowTable(fxnAhp)

tomdick-ahp-fxns

Here is the full YAML file for the “with preferenceFunction” case.

#########################
# Alternatives Section
# THIS IS FOR The Tom, Dick, & Harry problem at
# https://en.wikipedia.org/wiki/Analytic_hierarchy_process_%E2%80%93_leader_example
#
Alternatives: &alternatives
# 1= not well; 10 = best possible
# Your assessment based on the paragraph descriptions may be different.
  Tom:
    age: 50
    experience: 7
    education: 4
    leadership: 10
  Dick:
    age: 60
    experience: 10
    education: 6
    leadership: 6
  Harry:
    age: 30
    experience: 5
    education: 8
    leadership: 6
#
# End of Alternatives Section
#####################################
# Goal Section
#
Goal:
# A Goal HAS preferences (within-level comparison) and HAS Children (items in level)
  name: Choose the Most Suitable Leader
  preferences:
    # preferences are defined pairwise
    # 1 means: A is equal to B
    # 9 means: A is highly preferrable to B
    # 1/9 means: B is highly preferrable to A
    - [Experience, Education, 4]
    - [Experience, Charisma, 3]
    - [Experience, Age, 7]
    - [Education, Charisma, 1/3]
    - [Education, Age, 3]
    - [Age, Charisma, 1/5]
  children: 
    Experience:
          preferenceFunction: >
            ExperiencePreference <- function(a1, a2) {
              if (a1$experience < a2$experience) return (1/ExperiencePreference(a2, a1))
              ratio <- a1$experience / a2$experience
              if (ratio < 1.05) return (1)
              if (ratio < 1.2) return (2)
              if (ratio < 1.5) return (3)
              if (ratio < 1.8) return (4)
              if (ratio < 2.1) return (5) return (6) } children: *alternatives Education: preferenceFunction: >
            EducPreference <- function(a1, a2) {
              if (a1$education < a2$education) return (1/EducPreference(a2, a1))
              ratio <- a1$education / a2$education
              if (ratio < 1.05) return (1)
              if (ratio < 1.15) return (2)
              if (ratio < 1.25) return (3)
              if (ratio < 1.35) return (4)
              if (ratio < 1.55) return (5)
              return (5)
            }
          children: *alternatives
    Charisma:
      preferences:
        - [Tom, Dick, 5]
        - [Tom, Harry, 9]
        - [Dick, Harry, 4]
      children: *alternatives
    Age:
      preferences:
        - [Tom, Dick, 1/3]
        - [Tom, Harry, 5]
        - [Dick, Harry, 9]
      children: *alternatives
#
# End of Goal Section
#####################################

Analytic Hierarchy Process (AHP) with the ahp Package

On my December to-do list, I had “write an R package to make analytic hierarchy process (AHP) easier” — but fortunately gluc beat me to it, and saved me tons of time that I spent using AHP to do an actual research problem. First of all, thank you for writing the new ahp package! Next, I’d like to show everyone just how easy this package makes performing AHP and displaying the results. We will use the Tom, Dick, and Harry example that is described on Wikipedia. – the goal is to choose a new employee, and you can pick either Tom, Dick, or Harry. Read the problem statement on Wikipedia before proceeding.

AHP is a method for multi-criteria decision making that breaks the problem down based on decision criteria, subcriteria, and alternatives that could satisfy a particular goal. The criteria are compared to one another, the alternatives are compared to one another based on how well they comparatively satisfy the subcriteria, and then the subcriteria are examined in terms of how well they satisfy the higher-level criteria. The Tom-Dick-Harry problem is a simple hierarchy: only one level of criteria separates the goal (“Choose the Most Suitable Leader”) from the alternatives (Tom, Dick, or Harry):

tom-dick-harry

To use the ahp package, the most challenging part involves setting up the YAML file with your hierarchy and your rankings. THE MOST IMPORTANT THING TO REMEMBER IS THAT THE FIRST COLUMN IN WHICH A WORD APPEARS IS IMPORTANT. This feels like FORTRAN. YAML experts may be appalled that I just didn’t know this, but I didn’t. So most of the first 20 hours I spent stumbling through the ahp package involved coming to this very critical conclusion. The YAML AHP input file requires you to specify 1) the alternatives (along with some variables that describe the alternatives; I didn’t use them in this example, but I’ll post a second example that does use them) and 2) the goal hierarchy, which includes 2A) comparisons of all the criteria against one another FIRST, and then 2B) comparisons of the criteria against the alternatives. I saved my YAML file as tomdickharry.txt and put it in my C:/AHP/artifacts directory:

#########################
# Alternatives Section
# THIS IS FOR The Tom, Dick, & Harry problem at
# https://en.wikipedia.org/wiki/Analytic_hierarchy_process_%E2%80%93_leader_example
#
Alternatives: &alternatives
# 1= not well; 10 = best possible
# Your assessment based on the paragraph descriptions may be different.
  Tom:
    age: 50
    experience: 7
    education: 4
    leadership: 10
  Dick:
    age: 60
    experience: 10
    education: 6
    leadership: 6
  Harry:
    age: 30
    experience: 5
    education: 8
    leadership: 6
#
# End of Alternatives Section
#####################################
# Goal Section
#
Goal:
# A Goal HAS preferences (within-level comparison) and HAS Children (items in level)
  name: Choose the Most Suitable Leader
  preferences:
    # preferences are defined pairwise
    # 1 means: A is equal to B
    # 9 means: A is highly preferable to B
    # 1/9 means: B is highly preferable to A
    - [Experience, Education, 4]
    - [Experience, Charisma, 3]
    - [Experience, Age, 7]
    - [Education, Charisma, 1/3]
    - [Education, Age, 3]
    - [Age, Charisma, 1/5]
  children: 
    Experience:
      preferences:
        - [Tom, Dick, 1/4]
        - [Tom, Harry, 4]
        - [Dick, Harry, 9]
      children: *alternatives
    Education:
      preferences:
        - [Tom, Dick, 3]
        - [Tom, Harry, 1/5]
        - [Dick, Harry, 1/7]
      children: *alternatives
    Charisma:
      preferences:
        - [Tom, Dick, 5]
        - [Tom, Harry, 9]
        - [Dick, Harry, 4]
      children: *alternatives
    Age:
      preferences:
        - [Tom, Dick, 1/3]
        - [Tom, Harry, 5]
        - [Dick, Harry, 9]
      children: *alternatives
#
# End of Goal Section
#####################################

Next, I installed gluc’s ahp package and a helper package, data.tree, then loaded them into R:

devtools::install_github("gluc/ahp", build_vignettes = TRUE)
install.packages("data.tree")

library(ahp)
library(data.tree)

Running the calculations was ridiculously easy:

setwd("C:/AHP/artifacts")
myAhp <- LoadFile("tomdickharry.txt")
Calculate(myAhp)

And then generating the output was also ridiculously easy:

> GetDataFrame(myAhp)
                                  Weight  Dick   Tom Harry Consistency
1 Choose the Most Suitable Leader 100.0% 49.3% 35.8% 14.9%        4.4%
2  ¦--Experience                   54.8% 39.3% 11.9%  3.6%        3.2%
3  ¦--Education                    12.7%  1.0%  2.4%  9.2%        5.6%
4  ¦--Charisma                     27.0%  5.2% 20.1%  1.7%        6.1%
5  °--Age                           5.6%  3.8%  1.5%  0.4%        2.5%
> 
> print(myAhp, "weight", filterFun = isNotLeaf)
                        levelName     weight
1 Choose the Most Suitable Leader 1.00000000
2  ¦--Experience                  0.54756924
3  ¦--Education                   0.12655528
4  ¦--Charisma                    0.26994992
5  °--Age                         0.05592555
> print(myAhp, "weight")
                         levelName     weight
1  Choose the Most Suitable Leader 1.00000000
2   ¦--Experience                  0.54756924
3   ¦   ¦--Tom                     0.21716561
4   ¦   ¦--Dick                    0.71706504
5   ¦   °--Harry                   0.06576935
6   ¦--Education                   0.12655528
7   ¦   ¦--Tom                     0.18839410
8   ¦   ¦--Dick                    0.08096123
9   ¦   °--Harry                   0.73064467
10  ¦--Charisma                    0.26994992
11  ¦   ¦--Tom                     0.74286662
12  ¦   ¦--Dick                    0.19388163
13  ¦   °--Harry                   0.06325174
14  °--Age                         0.05592555
15      ¦--Tom                     0.26543334
16      ¦--Dick                    0.67162545
17      °--Harry                   0.06294121

You can also generate very beautiful output with the command below (but you’ll have to run the example yourself if you want to see how fantastically it turns out — maybe that will provide some motivation!)

ShowTable(myAhp)

I’ll post soon with an example of how to use AHP preference functions in the Tom, Dick, & Harry problem.

Course Materials for Statistics (The Easier Way) With R

very-quick-cover-outlineAre you an instructor with a Spring 2016 intro to statistics course coming up… and yet you haven’t started preparing? If so, I have a potential solution for you to consider. The materials (lecture slides, in-class practice problems in R, exams, syllabus) go with my book and are about 85% compiled, but good enough to get a course started this week (I will be finishing the collection by mid-January).
There is also a 36MB .zip file if you would like to download the materials.
Whether you will be using them or just considering them, please fill in the Google Form at https://docs.google.com/forms/d/1Z7djuKHg1L4k7bTtktHXI7Juduad3fUW9G69bxN6jRA/viewform so I can keep track of everyone and provide you with updates. Also, I want to make sure that I’m providing the materials to INSTRUCTORS (and not students), so please use the email account from your institution when you sign up. Once I get your contact information, I will email you the link to the materials.
If you would like permission to edit the materials, I can do that as well — I know a couple of you have expressed that you would like to add to the collection (e.g translate them to another language). If you see any issues or errors, please either fix and/or email to tell me to fix!
Thanks for your interest and participation! Also, Happy New Year!
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