The intercepts in the standardized solution

[Shu Fai Cheung (張樹輝)]

I would like to ask a quick question about the intercepts ("~1") in the standardized solution.

I understand that they are just the intercepts divided by the corresponding SDs:

https://groups.google.com/g/lavaan/c/kN5BRp4p_Wo/m/l__Th-lkBAAJ

A quick example:

``` r
library(lavaan)
#> This is lavaan 0.6-20
#> lavaan is FREE software! Please report any bugs.
dat <- HolzingerSwineford1939

mod <- "x1 ~ x4 + x7"

fit <- sem(mod,
           dat,
           fixed.x = FALSE,
           meanstructure = TRUE)
(est <- parameterEstimates(fit, standardized = TRUE))
#>   lhs op rhs   est    se      z pvalue ci.lower ci.upper std.lv std.all
#> 1  x1  ~  x4 0.373 0.054  6.855  0.000    0.267    0.480  0.373   0.372
#> 2  x1  ~  x7 0.002 0.058  0.039  0.969   -0.112    0.116  0.002   0.002
#> 3  x1 ~~  x1 1.170 0.095 12.268  0.000    0.983    1.357  1.170   0.861
#> 4  x4 ~~  x4 1.351 0.110 12.268  0.000    1.135    1.566  1.351   1.000
#> 5  x4 ~~  x7 0.220 0.074  2.971  0.003    0.075    0.365  0.220   0.174
#> 6  x7 ~~  x7 1.183 0.096 12.268  0.000    0.994    1.372  1.183   1.000
#> 7  x1 ~1     3.783 0.277 13.642  0.000    3.240    4.327  3.783   3.246
#> 8  x4 ~1     3.061 0.067 45.694  0.000    2.930    3.192  3.061   2.634
#> 9  x7 ~1     4.186 0.063 66.766  0.000    4.063    4.309  4.186   3.848
(implied_sd <- lavInspect(fit, "implied")$cov)
#>       x1    x4    x7
#> x1 1.358            
#> x4 0.505 1.351      
#> x7 0.085 0.220 1.183
est[7, "est"] / sqrt(implied_sd["x1", "x1"])
#> [1] 3.246046
est[7, "std.all"]
#> [1] 3.246046
```

This is also what Mplus does in the standardized solution.

This looks confusing because the "standardized solution" for the same simple model fitted by OLS will have the intercept equal to zero. However, given that both programs compute the intercept in the standardized solution in the same way, maybe there is a reason for doing so? What is this reason?

-- Shu Fai

[Felipe Vieira]

Hi Shu Fai,

If I understand your question w.r.t the reason, then my initial thought is that in this way we keep the model implied mean and covariance structure consistent after standardising it. We would "literally" be changing the model otherwise. 

Best, 

Felipe.