In reply to www.ljzigerell.com/?p=534 and his working paper here: www.ljzigerell.com/?p=2376

We are discussing his working paper over email, and I had some reservations about his factor analysis. I decided to run the analyses I wanted myself, but it turned into a longer project which should be placed in a short paper instead of in a private email.

I fetched the data from his source. The raw data did not have variable names, so was unwieldy to work with. I opened the SPSS file, and it did have variable names. Then I exported the CSV with the desired variables (see supp. material). Then I had to recoded the variables so that the true answers are coded as 1, false answers as 0, and missing as NA. This took some time. I followed his coding procedure for most cases (see his STATE file and my R code below).

How many factors to extract

It seems that he relies on some kind of method for determining the number of factors to extract, presumably Eigenvalue>1. I always use three different methods using the nFactors package. Using all 22 variables (note that he did not this all of them at once), all methods agreed to extract 5 factors (at max). Here’s the factor solutions for extracting 1 thru 5 factors and their intercorrelations:

Factor analyses with 1-5 factors and their correlations

[1] "Factor analysis, extracting 1 factors using oblimin and MinRes"

Loadings:
         MR1   
smokheal  0.129
condrift  0.347
rmanmade  0.445
earthhot  0.348
oxyplant  0.189
lasers    0.514
atomsize  0.441
antibiot  0.401
dinosaur  0.323
light     0.384
earthsun  0.515
suntime   0.581
dadgene   0.227
getdrug   0.290
whytest   0.423
probno4   0.396
problast  0.423
probreq   0.349
probif3   0.416
evolved   0.306
bigbang   0.315
onfaith  -0.296

                 MR1
SS loadings    3.191
Proportion Var 0.145
[1] "Factor analysis, extracting 2 factors using oblimin and MinRes"

Loadings:
         MR1    MR2   
smokheal  0.121       
condrift  0.345       
rmanmade  0.368  0.136
earthhot  0.363       
oxyplant  0.172       
lasers    0.518       
atomsize  0.461       
antibiot  0.323  0.133
dinosaur  0.323       
light     0.375       
earthsun  0.587       
suntime   0.658       
dadgene   0.145  0.130
getdrug   0.211  0.130
whytest   0.386       
probno4          0.705
problast         0.789
probreq   0.162  0.305
probif3   0.108  0.514
evolved   0.348       
bigbang   0.367       
onfaith  -0.266       

                 MR1   MR2
SS loadings    2.617 1.569
Proportion Var 0.119 0.071
Cumulative Var 0.119 0.190
     MR1  MR2
MR1 1.00 0.35
MR2 0.35 1.00
[1] "Factor analysis, extracting 3 factors using oblimin and MinRes"

Loadings:
         MR2    MR1    MR3   
smokheal                     
condrift                0.346
rmanmade  0.173  0.170  0.232
earthhot         0.187  0.220
oxyplant                0.100
lasers           0.256  0.320
atomsize         0.208  0.312
antibiot  0.168  0.150  0.198
dinosaur         0.119  0.250
light            0.240  0.169
earthsun         0.737       
suntime          0.754       
dadgene   0.147              
getdrug   0.152         0.149
whytest   0.108  0.143  0.294
probno4   0.708              
problast  0.781              
probreq   0.324              
probif3   0.532              
evolved                 0.562
bigbang                 0.525
onfaith                -0.307

                 MR2   MR1   MR3
SS loadings    1.646 1.444 1.389
Proportion Var 0.075 0.066 0.063
Cumulative Var 0.075 0.140 0.204
     MR2  MR1  MR3
MR2 1.00 0.29 0.25
MR1 0.29 1.00 0.43
MR3 0.25 0.43 1.00
[1] "Factor analysis, extracting 4 factors using oblimin and MinRes"

Loadings:
         MR4    MR2    MR1    MR3   
smokheal                            
condrift  0.180                0.234
rmanmade  0.387                     
earthhot  0.262         0.102       
oxyplant  0.116                     
lasers    0.490                     
atomsize  0.435                     
antibiot  0.485                     
dinosaur  0.312                     
light     0.274         0.142       
earthsun                0.797       
suntime                 0.719       
dadgene   0.234                     
getdrug   0.273                     
whytest   0.438                     
probno4          0.695              
problast         0.817              
probreq   0.180  0.275              
probif3   0.139  0.487              
evolved                        0.685
bigbang                        0.554
onfaith  -0.141               -0.230

                 MR4   MR2   MR1   MR3
SS loadings    1.511 1.501 1.204 0.915
Proportion Var 0.069 0.068 0.055 0.042
Cumulative Var 0.069 0.137 0.192 0.233
     MR4  MR2  MR1  MR3
MR4 1.00 0.39 0.57 0.42
MR2 0.39 1.00 0.23 0.12
MR1 0.57 0.23 1.00 0.27
MR3 0.42 0.12 0.27 1.00
[1] "Factor analysis, extracting 5 factors using oblimin and MinRes"

Loadings:
         MR2    MR1    MR3    MR5    MR4   
smokheal                                   
condrift                0.209         0.299
rmanmade  0.104                0.120  0.379
earthhot                              0.367
oxyplant                              0.220
lasers                         0.195  0.361
atomsize                       0.273  0.207
antibiot                       0.401  0.108
dinosaur                       0.204  0.131
light                                 0.423
earthsun         0.504                0.186
suntime          1.007                     
dadgene                        0.277       
getdrug                        0.373       
whytest                        0.504       
probno4   0.701                            
problast  0.816                            
probreq   0.272                0.174       
probif3   0.487                0.107       
evolved                 0.753              
bigbang                 0.483         0.165
onfaith                -0.225 -0.152       

                 MR2   MR1   MR3   MR5   MR4
SS loadings    1.501 1.291 0.919 0.874 0.871
Proportion Var 0.068 0.059 0.042 0.040 0.040
Cumulative Var 0.068 0.127 0.169 0.208 0.248
     MR2  MR1  MR3  MR5  MR4
MR2 1.00 0.20 0.11 0.38 0.28
MR1 0.20 1.00 0.21 0.41 0.44
MR3 0.11 0.21 1.00 0.32 0.30
MR5 0.38 0.41 0.32 1.00 0.50
MR4 0.28 0.44 0.30 0.50 1.00

Interpretation

We see that in the 1-factor solution, all variables load in the expected direction, and we can speak of a general scientific knowledge factor. This is the one we want to use for other analyses. We see that faith loads negatively. This variable is not a true/false question, and thus should be excluded from any actual measurement of the general scientific knowledge factor.

Increasing the number of factors to extract simply divides this general factor into correlated parts. E.g. in the 2-factor solution, we see a probability factor that correlates .35 with the remaining semi-general factor. In solution 3, we see MR2 as the probability factor, MR3 as the knowledge related to religious beliefs factor and MR1 as the remaining items. Intercorrelations are .29, .25 and .43. This pattern continues until the 5th solution which still produces 5 correlated factors: MR2 is the probability factor, MR1 is an astronomy factor, MR3 is the one having to do with religious beliefs, MR5 looks like a medicine/genetics factor, and MR4 is the rest.

Just because scree tests etc. tell you to extract >1 factor does not mean that there is no general factor. This is the old fallacy made in the study of cognitive ability. See discussion in Jensen 1998 (chapter 3). It is sometimes still made e.g. Hampshire, et al (2012). Generally, as one increases the number of variables, the suggested number of factors to extract goes up. This does not mean that there is no general factor, just that with increasing number of variables, one can see a more fine-grained structure in the data than one can with only e.g. 5 variables.

Should we use them or not?

Before discussing whether one should theoretically use them or not, one can measure if it makes much of a difference. One can do this by extracting the general factor with and without the items in questions. I did this, also excluding the onfaith item. Then I correlated the scores from these two analysis: r=.992. In other words, it hardly matters whether one includes these religious-tinged items or not. The general factor is measured quite well already without them and they do not substantially change the factor scores. However, since adding more indicator items/variables generally reduces measurement error of a latent trait/factor, I would include them in my analyses.

How many factors should we extract and use?

There is also the question of how many factors one should extract. The answer is that it depends on what one wants to do. As Zigerell points out in a review comment of this paper on Winnower:

For example, for diagnostic purposes, if we know only that students A, B, and C miss 3 items on a test of general science knowledge, then the only remediation is more science; but we can provide more tailored remediation if we have separate components so that we observe that, say, A did poorly only on the religion-tinged items, B did poorly only on the probability items, and C did poorly only on the astronomy items.

For remedial education, it is clearly preferable to extract the highest number of interpretable factors because this gives the most precise information where knowledge is lacking for a given person. In regression analysis where we want to control for scientific knowledge, one should use the general factor.

References

Hampshire, A., Highfield, R. R., Parkin, B. L., & Owen, A. M. (2012). Fractionating human intelligence. Neuron, 76(6), 1225-1237.

Jensen, A. R. (1998). The g factor: The science of mental ability. Westport, CT: Praeger.

Supplementary material

Datafile: science_data

R code

library(plyr) #for mapvalues

data = read.csv("science_data.csv") #load data

#Coding so that 1 = true, 0 = false
data$smokheal = mapvalues(data$smokheal, c(9,7,8,2),c(NA,0,0,0))
data$condrift = mapvalues(data$condrift, c(9,7,8,2),c(NA,0,0,0))
data$earthhot = mapvalues(data$earthhot, c(9,7,8,2),c(NA,0,0,0))
data$rmanmade = mapvalues(data$rmanmade, c(9,7,8,1,2),c(NA,0,0,0,1)) #reverse
data$oxyplant = mapvalues(data$oxyplant, c(9,7,8,2),c(NA,0,0,0))
data$lasers = mapvalues(data$lasers, c(9,7,8,2,1),c(NA,0,0,1,0)) #reverse
data$atomsize = mapvalues(data$atomsize, c(9,7,8,2),c(NA,0,0,0))
data$antibiot = mapvalues(data$antibiot, c(9,7,8,2,1),c(NA,0,0,1,0)) #reverse
data$dinosaur = mapvalues(data$dinosaur, c(9,7,8,2,1),c(NA,0,0,1,0)) #reverse
data$light = mapvalues(data$light, c(9,7,8,2,3),c(NA,0,0,0,0))
data$earthsun = mapvalues(data$earthsun, c(9,7,8,2),c(NA,0,0,0))
data$suntime = mapvalues(data$suntime, c(9,7,8,2,3,1,4,99),c(0,0,0,0,1,0,0,NA))
data$dadgene = mapvalues(data$dadgene, c(9,7,8,2),c(NA,0,0,0))
data$getdrug = mapvalues(data$getdrug, c(9,7,8,2,1),c(NA,0,0,1,0)) #reverse
data$whytest = mapvalues(data$whytest, c(1,2,3,4,5,6,7,8,9,99),c(1,0,0,0,0,0,0,0,0,NA))
data$probno4 = mapvalues(data$probno4, c(9,8,2,1),c(NA,0,1,0)) #reverse
data$problast = mapvalues(data$problast, c(9,8,2,1),c(NA,0,1,0)) #reverse
data$probreq = mapvalues(data$probreq, c(9,8,2),c(NA,0,0))
data$probif3 = mapvalues(data$probif3, c(9,8,2,1),c(NA,0,1,0)) #reverse
data$evolved = mapvalues(data$evolved, c(9,7,8,2),c(NA,0,0,0))
data$bigbang = mapvalues(data$bigbang, c(9,7,8,2),c(NA,0,0,0))
data$onfaith = mapvalues(data$onfaith, c(9,1,2,3,4,7,8),c(NA,1,1,0,0,0,0))

#How many factors to extract?
library(nFactors)
nScree(data[complete.cases(data),]) #use complete cases only

#extract factors
library(psych) #for factor analysis
for (num in 1:5) {
  print(paste0("Factor analysis, extracting ",num," factors using oblimin and MinRes"))
  fa = fa(data,num) #extract factors
  print(fa$loadings) #print
  if (num>1){ #print factor cors
    phi = round(fa$Phi,2) #round to 2 decimals
    colnames(phi) = rownames(phi) = colnames(fa$scores) #set names
    print(phi) #print
  }
}

#Does it make a difference?
fa.all = fa(data[1:21]) #no onfaith
fa.noreligious = fa(data[1:19]) #no onfaith, bigbang, evolved
cor(fa.all$scores,fa.noreligious$scores, use="pair") #correlation, ignore missing cases

So I was installing something (forgot what), and had problems with RCurl:

* installing *source* package ‘RCurl’ …
** package ‘RCurl’ successfully unpacked and MD5 sums checked
checking for curl-config… no
Cannot find curl-config
ERROR: configuration failed for package ‘RCurl’
* removing ‘/home/lenovo/R/x86_64-pc-linux-gnu-library/3.1/RCurl’
Warning in install.packages :
installation of package ‘RCurl’ had non-zero exit status

Solution was to do sudo apt-get install libcurl4-openssl-dev (found in a comment here).

Getting a percentage table from a dataframe

A reviewer asked me to:

1) As I said earlier, there should be some data on the countries of origin of the immigrant population. Most readers have no idea who actually moves to Denmark. At the very least, there should be basic information like “x% of the immigrant population is of non-European origin and y% of European origin as of 2014.” Generally, non-European immigration would be expected to increase inequality more, given that IQ levels are relatively uniform across Europe.

I have population counts for each year 1980 through 2014 in a dataframe and I’d like to get them as a percent of each year so as to get the relative sizes of the countries. There is a premade function for this, prop.table, however, it works quite strangely. If one gives it a dataframe and no margin, it will use the total sum of the data.frame instead of by column. This is sometimes useful, but not in this case. However, if one gives it a data.frame and margin=2, it will complain that:

Error in margin.table(x, margin) : 'x' is not an array

Which is odd when it just accepted it before. The relatively lack of documentation made it not quite easy to figure out how to make it work. Turns out that one just has to convert the dataframe to a matrix when giving it:

census.percent = prop.table(as.matrix(census), margin=2)

and then one can convert it back and also multiple by 100 to get percent instead of fractions:

census.percent = as.data.frame(prop.table(as.matrix(census), margin=2)*100)

Getting the top 10 countries with names for selected years

This one was harder. Here’s the code I ended up with:

selected.years = c("X1980","X1990","X2000","X2010","X2014") #years of interest
for (year in selected.years){ #loop over each year of interest
  vector = census.percent[,year,drop=FALSE] #get the vector, DONT DROP!
  View(round(vector[order(vector, decreasing = TRUE),,drop=FALSE][1:10,,drop=FALSE],1)) #sort vector, DONT drop! and get 1:10 and DONT DROP!
}

First we choose the years we want (note that X goes in front because R has trouble handling columns that begin with a number). Then we loop over each year of interest. Then we pick it out to avoid having to select the same column over and over. However, normally when picking out 1 column from a dataframe, R will convert it to numeric, which is very bad because this removes the rownames. That means that even tho we can find the top 10 countries, we don’t know which ones they are. The solution for this is to set drop=FALSE. The next part consists of first ordering the vector (without drop!), and then selecting the top 10 countries without dropping. I open them in View (in Rstudio) because this makes it easier to copy the values for further use (e.g. in a table for a paper).

So, drop=FALSE is another one of those pesky small things to remember. It is just like stringsAsFactors=FALSE when using read.table (or read.csv).

 

For a mathematical explanation of the test, see e.g. here. However, such an explanation is not very useful for using the test in practice. Just what does a W value of .95 mean? What about .90 or .99? One way to get a feel for it, is to simulate datasets, plot them and calculate the W values. Additionally, one can check the sensitivity of the test, i.e. the p value.

All the code is in R.

#random numbers from normal distribution
set.seed(42) #for reproducible numbers
x = rnorm(5000) #generate random numbers from normal dist
hist(x,breaks=50, main="Normal distribution, N=5000") #plot
shapiro.test(x) #SW test
>W = 0.9997, p-value = 0.744

SW_norm

So, as expected, W was very close to 1, and p was large. In other words, SW did not reject a normal distribution just because N is large. But maybe it was a freak accident. What if we were to repeat this experiment 1000 times?

#repeat sampling + test 1000 times
Ws = numeric(); Ps = numeric() #empty vectors
for (n in 1:1000){ #number of simulations
  x = rnorm(5000) #generate random numbers from normal dist
  sw = shapiro.test(x)
  Ws = c(Ws,sw$statistic)
  Ps = c(Ps,sw$p.value)
}
hist(Ws,breaks=50) #plot W distribution
hist(Ps,breaks=50) #plot P distribution
sum(Ps<.05) #how many Ps below .05?

The number of Ps below .05 was in fact 43, or 4.3%. I ran the code with 100,000 simulations too, which takes 10 minutes or something. The value was 4389, i.e. 4.4%. So it seems that the method used to estimate the P value is slightly off in that the false positive rate is lower than expected.

What about the W statistic? Is it sensitive to fairly small deviations from normality?

#random numbers from normal distribution, slight deviation
x = c(rnorm(4900),rnorm(100,2))
hist(x,breaks=50, main="Normal distribution N=4900 + normal distribution N=200, mean=2")
shapiro.test(x)
>W = 0.9965, p-value = 1.484e-09


Here I started with a very large norm. dist. and added a small norm dist. to it with a different mean. The difference is hardly visible to the eye, but the P value is very small. The reason is that the large sample size makes it possible to detect even very small deviations from normality. W was again very close to 1, indicating that the distribution was close to normal.

What about a decidedly non-normal distribution?

#random numbers between -10 and 10
x = runif(5000, min=-10, max=10)
hist(x,breaks=50,main="evenly distributed numbers [-10;10], N=5000")
shapiro.test(x)
>W = 0.9541, p-value < 2.2e-16

 

SW_even

SW wisely rejects this with great certainty as being normal. However, W is near 1 still (.95). This tells us that the W value does not vary very much even when the distribution is decidedly non-normal. For interpretation then, we should probably bark when W drops just under .99 or so.

As a further test of the W values, here’s two equal sized distributions plotted together.

#normal distributions, 2 sd apart (unimodal fat normal distribution)
x = c(rnorm(2500, -1, 1),rnorm(2500, 1, 1))
hist(x,breaks=50,main="Mormal distributions, 2 sd apart")
shapiro.test(x)
>W = 0.9957, p-value = 6.816e-11
sd(x)
>1.436026

SW_norm3 It still looks fairly normal, altho too fat. The standard deviation is in fact 1.44, or 44% larger than it is supposed to be. The W value is still fairly close to 1, however, and only a little less than from the distribution that was only slightly nonnormal (Ws = .9957 and .9965). What about clearly bimodal distributions?

#bimodal normal distributions, 4 sd apart
x = c(rnorm(2500, -2, 1),rnorm(2500, 2, 1))
hist(x,breaks=50,main="Normal distributions, 4 sd apart")
shapiro.test(x)
>W = 0.9464, p-value < 2.2e-16

SW_norm4

This clearly looks nonnormal. SW rejects it rightly and W is about .95 (W=0.9464). This is a bit lower than for the evenly distributed numbers. (W=0.9541)

What about an extreme case of nonnormality?

#bimodal normal distributions, 20 sd apart
x = c(rnorm(2500, -10, 1),rnorm(2500, 10, 1))
hist(x,breaks=50,main="Normal distributions, 20 sd apart")
shapiro.test(x)
>W = 0.7248, p-value < 2.2e-16

SW_norm5

Finally we make a big reduction in the W value.

What about some more moderate deviations from normality?

#random numbers from normal distribution, moderate deviation
x = c(rnorm(4500),rnorm(500,2))
hist(x,breaks=50, main="Normal distribution N=4500 + normal distribution N=500, mean=2")
shapiro.test(x)
>W = 0.9934, p-value = 1.646e-14

SW_norm6

This one has a longer tail on the right side, but it still looks fairly normal. W=.9934.

#random numbers from normal distribution, large deviation
x = c(rnorm(4000),rnorm(1000,2))
hist(x,breaks=50, main="Normal distribution N=4000 + normal distribution N=1000, mean=2")
shapiro.test(x)
>W = 0.991, p-value < 2.2e-16

SW_norm7

This one has a very long right tail. W=.991.

In conclusion

Generally we see that given a large sample, SW is sensitive to departures from non-normality. If the departure is very small, however, it is not very important.

We also see that it is hard to reduce the W value even if one deliberately tries. One needs to test extremely non-normal distribution in order for it to fall appreciatively below .99.

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arstechnica.com/gaming/2013/10/when-a-virtual-actress-nude-images-leak-who-should-take-the-legal-blame/

 

When nude images of Jodie Holmes, actress Ellen Page’s character from Beyond: Two Souls, began appearing on the Internet a few weeks ago (courtesy of a repositioned shower-scene camera running on debug hardware) we thought the story was a little too tabloidy to cover. This kind of embarrassing, tawdry celebrity gossip is pretty common in the entertainment industry, even if it’s relatively rare in video games particularly. Scandals revolving around supposedly inaccessible adult content in games aren’t completely unheard of, though; remember GTA: San AndreasHot Coffee?

But when reports surfaced earlier this week that Sony was making vague legal threats in an effort to remove those images from the Internet, our news ears started perking up a little.

Nordic entertainment site Eskimo Press was the first to report that Sony Computer Entertainment Europe asked them to take down the leaked images, citing unspecified “legal reasons” for the request. This action came despite the fact that Eskimo Press merely linked to the images on another server rather than hosting them itself. Culture site Gaming Blend said it received a similar request from Sony Computer Entertainment America, which went so far as to request that the original story be taken down entirely.

 

I hate censorship. I went to /b/ to get the pics. They are nothing special.

Technically, these shudnt apply in DK anyway, or what?

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Introducing the construct curiosity for predicting job performance

Mussel, Patrick. “Introducing the construct curiosity for predicting job performance.” Journal of Organizational Behavior (2012).
I read this a while ago. pretty interesting.

Summary

The present paper provides a conceptual and empirical examination regarding the relevance of the construct curiosity for work-related outcomes. On the basis of a review and integration of the literature regarding the construct itself, the construct is conceptually linked with performance in the work context. In line with a confirmatory research strategy, the sample of the present study (N = 320) has requirements which reflect this conceptual link. Results from a concurrent validation study confirmed the hypothesis regarding the significance of curiosity for job performance (r = .34). Furthermore, incremental validity of curiosity above 12 cognitive and non-cognitive predictors for job performance suggests that curiosity captures variance in the criterion that is not explained by predictors traditionally used in organizational psychology. It is concluded that curiosity is an important variable for the prediction and explanation of work-related behavior. Furthermore, given the dramatic changes in the world of work, the importance is likely to rise, rather than to decline, which has important implications for organizational theories and applied purposes, such as personnel selection.