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

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

MR1
smokheal  0.129
condrift  0.347
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
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
Proportion Var 0.145
[1] "Factor analysis, extracting 2 factors using oblimin and MinRes"

MR1    MR2
smokheal  0.121
condrift  0.345
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
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
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"

MR2    MR1    MR3
smokheal
condrift                0.346
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
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
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"

MR4    MR2    MR1    MR3
smokheal
condrift  0.180                0.234
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
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
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"

MR2    MR1    MR3    MR5    MR4
smokheal
condrift                0.209         0.299
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
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
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

#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\$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\$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