# How can I interpret having vastly different Pr(>|t|) values from a seemingly minor change in a linear regression model for prediction?

I have two models which will be used for prediction. The predictor variable zq65 has very different summary results for Pr(>|t|), depending on the inclusion of an interaction term in the model.

Is this unusual to have Pr(>|t|) change so much (0.969479 vs. 0.06721), with the addition of another predictor (i.e., I(zmean*zpcum5))?

zpcum5 has a negative (-0.552) correlation with the data, so maybe I shouldn't be combining this with zmean which has a positive (0.800) correlation? As you can see, I'm not sure how to proceed.

# -----------------------------------------------------------------------------------------------
fmla_sqrtf  <- as.formula("plotVol_sqrt   ~ zmean + zq65 + zpcum5 + I(zmean*zpcum5) + mc20210624 * mc20210425")
fmla_sqrtj  <- as.formula("plotVol_sqrt   ~ zmean + zq65 + zpcum5                   + mc20210624 * mc20210425")

Lm2_sqrtf <- lm(fmla_sqrtf, data = lidarDataSubset_B)
Lm2_sqrtj <- lm(fmla_sqrtj, data = lidarDataSubset_B)
# -----------------------------------------------------------------------------------------------

# -----------------------------------------------------------------------------------------------
# summary result for Lm2_sqrtf
# -----------------------------------------------------------------------------------------------
> summary(Lm2_sqrtf)

Call:
lm(formula = fmla_sqrtf, data = lidarDataSubset_B)

Residuals:
Min      1Q  Median      3Q     Max
-4.7570 -1.1558  0.1995  1.2259  4.9739

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)           -29.373843  18.350664  -1.601 0.110469
zmean                   0.992839   0.278693   3.562 0.000426 ***
zq65                    0.009464   0.247136   0.038 0.969479
zpcum5                 -0.054502   0.015291  -3.564 0.000423 ***
I(zmean * zpcum5)       0.006974   0.002002   3.483 0.000568 ***
mc20210624             27.012833  20.034941   1.348 0.178557
mc20210425             64.169975  27.869473   2.303 0.021972 *
mc20210624:mc20210425 -59.130015  30.287534  -1.952 0.051810 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.807 on 308 degrees of freedom
Multiple R-squared:  0.7886,    Adjusted R-squared:  0.7838
F-statistic: 164.2 on 7 and 308 DF,  p-value: < 2.2e-16
# -----------------------------------------------------------------------------------------------

# -----------------------------------------------------------------------------------------------
# summary result for Lm2_sqrtj
# -----------------------------------------------------------------------------------------------
> summary(Lm2_sqrtj)

Call:
lm(formula = fmla_sqrtj, data = lidarDataSubset_B)

Residuals:
Min      1Q  Median      3Q     Max
-4.6753 -1.1852  0.1644  1.2388  5.2397

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)           -24.777054  18.629852  -1.330  0.18451
zmean                   0.772571   0.276266   2.796  0.00549 **
zq65                    0.409188   0.222777   1.837  0.06721 .
zpcum5                 -0.007396   0.007260  -1.019  0.30912
mc20210624             19.686892  20.279864   0.971  0.33243
mc20210425             53.435600  28.192944   1.895  0.05898 .
mc20210624:mc20210425 -46.913907  30.620744  -1.532  0.12652
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.839 on 309 degrees of freedom
Multiple R-squared:  0.7803,    Adjusted R-squared:  0.776
F-statistic: 182.9 on 6 and 309 DF,  p-value: < 2.2e-16
# -----------------------------------------------------------------------------------------------

• "zpcum5 has a negative (-0.552) correlation with the data" - "the data" means what? "(0.969479 vs. 0.06721), with the addition of another predictor (i.e., I(zmean zpcum5))?" If I see things correctly, the larger p-value is with the interaction and the smaller without, right? Commented Jan 17, 2023 at 17:19
• Yes, it is usual. That's the main point of considering interactions: they can make a difference.
– whuber
Commented Jan 17, 2023 at 17:33
• @ChristianHennig ... the data is volume (square root actually) for a forest plot. yes, the larger p-value is the model that has the interaction. Ideally, I would want to use zmean, zq65, and zpcum5 because each represent a different 'dimension' of the lidar's metrics in a forest plot. But it looks like, for prediction, it's better to drop one. Commented Jan 17, 2023 at 17:41
• okay thanks @whuber. I think, for this data, the interaction is important so would take precedence. The big change threw me, but I guess it just means that even though zq65 has a correlation with volume of 0.800 as well, it doesn't contribute to the first model. Commented Jan 17, 2023 at 17:51
• You could do some cross-validation to compare the prediction performance of the models. Commented Jan 17, 2023 at 17:53