Regression analysis with discrete dependent variable and continuous independent var.

[blindfreak]

Dear fellow users,
I am confronted with the following task:

I want to regress multiple (5) continuous independent variables against a discrete independent variable. Is a linear regression applicable? If not, how can I do that with minitab?

Some more details:
My dependent variable is the output of an 'ideal employer' ranking and ranges from 1 to 50.
My independent variables are three continuous variables that measure corporate environmental performance (ranging from 1,0 to 100) and two continuous variables that capture corporate revenue and profits (my control variables).

If more information is required, please let me know.

Many thanks!!!
Linus

[Miner]

I think you are trying to say that you have 5 continuous independent (X) variables and a discrete (ordinal) dependent (Y) response variable.

You have two options:

1. Ordinal logistic regression - this is designed specifically for your situation.
2. Multiple regression - Discrete variables like count data can behave as a continuous variable if there are enough levels. While ordinal data is not exactly the same (i.e., spacing between levels is not assured), the fact that your scale has 50 levels leads me to think this might be a viable option. You could test this by taking a sample of the ranking data, generating a histogram and visually comparing it to a normal distribution. For example, if you generate a Poisson distribution for counts > 10, it will approximate the normal distribution. You can then use the Freeman-Tukey transform for count data and analyze it as if it were continuous data.

[blindfreak]

Dear Miner,

First of all, thank you sooo much for your quick and helpful reply! I deeply appraciate it!

Unfortunately, I am an untaught amateur and that is why I am not too sure whether ordinal logistic regression does indeed fit. Is it really applicable if my ordinal dependent response variable does count up from 1 to 71 (in fact, my sample size is 71) and each ranking/number does occur only once?

I will post my data set below this message, perhaps this eases the understanding of my problem.

It is hilarious!!! I am trying for half an hour right now to insert a table --> without success!!!


This is the minitab output:

Logistic Regression Table
Odds 95% CI
Predictor Coef SE Coef Z P Ratio Lower Upper
Const(1) -8,04078 1,87759 -4,28 0,000
Const(2) -7,32818 1,73513 -4,22 0,000
Const(3) -6,90195 1,68333 -4,10 0,000
Const(4) -6,59339 1,65571 -3,98 0,000
Const(5) -6,34926 1,63805 -3,88 0,000
Const(6) -6,14476 1,62542 -3,78 0,000
Const(7) -5,96395 1,61553 -3,69 0,000
Const(8) -5,79643 1,60722 -3,61 0,000
Const(9) -5,64020 1,60006 -3,52 0,000
Const(10) -5,49793 1,59395 -3,45 0,001
Const(11) -5,36386 1,58848 -3,38 0,001
Const(12) -5,23538 1,58345 -3,31 0,001
Const(13) -5,11773 1,57900 -3,24 0,001
Const(14) -5,00979 1,57503 -3,18 0,001
Const(15) -4,90784 1,57137 -3,12 0,002
Const(16) -4,80961 1,56790 -3,07 0,002
Const(17) -4,71624 1,56466 -3,01 0,003
Const(18) -4,62791 1,56164 -2,96 0,003
Const(19) -4,54085 1,55870 -2,91 0,004
Const(20) -4,45480 1,55583 -2,86 0,004
Const(21) -4,37089 1,55306 -2,81 0,005
Const(22) -4,28585 1,55028 -2,76 0,006
Const(23) -4,20059 1,54752 -2,71 0,007
Const(24) -4,11800 1,54488 -2,67 0,008
Const(25) -4,03727 1,54232 -2,62 0,009
Const(26) -3,95953 1,53988 -2,57 0,010
Const(27) -3,88578 1,53758 -2,53 0,011
Const(28) -3,81350 1,53536 -2,48 0,013
Const(29) -3,74262 1,53320 -2,44 0,015
Const(30) -3,67396 1,53112 -2,40 0,016
Const(31) -3,60678 1,52912 -2,36 0,018
Const(32) -3,53937 1,52713 -2,32 0,020
Const(33) -3,47254 1,52517 -2,28 0,023
Const(34) -3,40648 1,52327 -2,24 0,025
Const(35) -3,33904 1,52135 -2,19 0,028
Const(36) -3,27022 1,51942 -2,15 0,031
Const(37) -3,20181 1,51753 -2,11 0,035
Const(38) -3,13526 1,51572 -2,07 0,039
Const(39) -3,06974 1,51398 -2,03 0,043
Const(40) -3,00455 1,51227 -1,99 0,047
Const(41) -2,93988 1,51062 -1,95 0,052
Const(42) -2,87495 1,50900 -1,91 0,057
Const(43) -2,80905 1,50739 -1,86 0,062
Const(44) -2,74360 1,50584 -1,82 0,068
Const(45) -2,67804 1,50434 -1,78 0,075
Const(46) -2,61067 1,50285 -1,74 0,082
Const(47) -2,54179 1,50138 -1,69 0,090
Const(48) -2,47120 1,49994 -1,65 0,099
Const(49) -2,39832 1,49854 -1,60 0,110
Const(50) -2,32299 1,49717 -1,55 0,121
Const(51) -2,24607 1,49587 -1,50 0,133
Const(52) -2,16760 1,49465 -1,45 0,147
Const(53) -2,08722 1,49352 -1,40 0,162
Const(54) -2,00464 1,49250 -1,34 0,179
Const(55) -1,91967 1,49161 -1,29 0,198
Const(56) -1,83068 1,49085 -1,23 0,219
Const(57) -1,73587 1,49027 -1,16 0,244
Const(58) -1,63887 1,48992 -1,10 0,271
Const(59) -1,53861 1,48985 -1,03 0,302
Const(60) -1,43265 1,49011 -0,96 0,336
Const(61) -1,32030 1,49081 -0,89 0,376
Const(62) -1,19695 1,49212 -0,80 0,422
Const(63) -1,05993 1,49429 -0,71 0,478
Const(64) -0,907912 1,49768 -0,61 0,544
Const(65) -0,737512 1,50287 -0,49 0,624
Const(66) -0,541897 1,51086 -0,36 0,720
Const(67) -0,306970 1,52382 -0,20 0,840
Const(68) -0,0051660 1,54697 -0,00 0,997
Const(69) 0,413806 1,59464 0,26 0,795
Const(70) 1,11971 1,73447 0,65 0,519
profits $ mio 0,0000535 0,0000332 1,61 0,107 1,00 1,00 1,00
revenue $ mio 0,0000009 0,0000029 0,30 0,762 1,00 1,00 1,00
DISCLOSURE -0,0043659 0,0086872 -0,50 0,615 1,00 0,98 1,01
ENVTL. MGMT. -0,0006438 0,0177172 -0,04 0,971 1,00 0,97 1,03
ENVTL. IMPACT 0,0544707 0,0179416 3,04 0,002 1,06 1,02 1,09
　
Log-Likelihood = -294,794
Test that all slopes are zero: G = 15,712, DF = 5, P-Value = 0,008
　
Goodness-of-Fit Tests
Method Chi-Square DF P
Pearson 4886,37 4895 0,532
Deviance 589,59 4895 1,000
　
Measures of Association:
(Between the Response Variable and Predicted Probabilities)
Pairs Number Percent Summary Measures
Concordant 1625 65,4 Somers' D 0,31
Discordant 849 34,2 Goodman-Kruskal Gamma 0,31
Ties 11 0,4 Kendall's Tau-a 0,31
Total 2485 100,0


How do I interpret the results? Do my independet variables determine my dependent variable? How strong is the relationship?


I know that this is very detailed and that I cannot expect anyone to invest his time in my problems, but I hope that some expert can help me with this

Sincere thanks!
Linus

[Miner]

Are you ranking individuals relative to each other, or rating them against a standard?

[blindfreak]

I am using Newsweek's Ideal Employer Ranking, i.e. students ranked their most favorite companies to work for. E.g. Apple is ranked number 1, China Mobile number 2, Google on 3, and so on.
The H0 hypothesis is that corporate environmental performance does not determine perceived company attractiveness, i.e. I want to show that objective environmental performance values (my independent continuous variables) do have an influence on corporate attractiveness ratings (my dependent variable).

[Miner]

I am going to have to think on this. Since the response is ranked data, it violates the independence assumptions of your standard regression approaches.

Disregard my earlier options, ordinal and multiple regression would not be appropriate for this scenario.

[Steve Prevette]

Unfortunately, what you really need is the measurement data that were behind how Newsweek came up with the ranking. You don't know if 1st and 2nd were real close together, but a wide gulf between 2nd and 3rd. Without the continuous source data behind the ranking, you are at a real disadvantage. I doubt Newsweek would release the data, but you could always try contacting Newsweek and/or the authors.

[Miner]

Steve makes good comments. I researched this and the appropriate technique is ranked regression. Unfortunately, Minitab does not perform ranked regression, or even Spearman's ranked correlation. If you were only dealing with a single predictor, I could approximate it, but not with five predictors.