Measuring scientific knowledge: can we use questions that are denied by the religious?

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
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