From this post, and making reference to this other post, this, and page 9 of this.

The formula for the standardized residuals is:

\[\text{Pearson's residuals}\,=\,\frac{\text{Observed - Expected}}{ \sqrt{\text{Expected}}}\]

the sum of squared standardized residuals is the chi square value.

Assuming a level of significance of \(0.05\), the cutoff limit for statistical significance is \(\pm1.96\), or an absolute value greater than \(1.96.\)

The fact that this is a three-way contingency table complicates the interpretation, which is very nicely explained in @roando2’s answer.

Here is a simulation with a made-up table that resembles the OP to clarify the calculations:

```
tab_df = data.frame(expand.grid(
age = c("15-24", "25-39", ">40"),
attitude = c("no","moderate"),
memory = c("yes", "no")),
count = c(1,4,3,1,8,39,32,36,25,35,32,38) )
(tab = xtabs(count ~ ., data = tab_df))
, , memory = yes
attitude
age no moderate
15-24 1 1
25-39 4 8
>40 3 39
, , memory = no
attitude
age no moderate
15-24 32 35
25-39 36 32
>40 25 38
require(vcd)
mosaic(~ memory + age + attitude, data = tab, shade = T)
expected = mosaic(~ memory + age + attitude, data = tab, type = "expected")
expected
# Finding, as an example, the expected counts in >40 with memory and moderate att.:
over_forty = sum(3,39,25,38)
mem_yes = sum(1,4,3,1,8,39)
att_mod = sum(1,8,39,35,32,38)
exp_older_mem_mod = over_forty * mem_yes * att_mod / sum(tab)^2
# Corresponding standardized Pearson's residual:
(39 - exp_older_mem_mod) / sqrt(exp_older_mem_mod) # [1] 6.709703
```

It is interesting to compare the graphical representation to the results of the Poisson regression, which illustrates perfectly the English interpretation in @rolando2 ’s answer:

fit <- glm(count ~ age + attitude + memory, data=tab_df, family=poisson()) summary(fit)

```
Call:
glm(formula = count ~ age + attitude + memory, family = poisson(),
data = tab_df)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.4491 -1.8546 -1.0853 0.8647 5.4873
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.7999 0.1854 9.708 < 2e-16 ***
age25-39 0.1479 0.1643 0.900 0.36794
age>40 0.4199 0.1550 2.709 0.00674 **
attitudemoderate 0.4153 0.1282 3.239 0.00120 **
memoryno 1.2629 0.1514 8.344 < 2e-16 ***
```