Adjust 95% CI and P value if I scale up predictors in the logistic regression? I am conducting a logistic regression in R and trying to understand whether to adjust the p-value and 95% CI, in the same way, I did for the coefficient/odds ratio if I want to report "every 5-units increase" in BMI (pls see the code below)? If so, how can I do that?
model <- glm(disease~ BMI, data=mydata, family=binomial)
exp(model$coefficients*5) ## odds ratio for each 5-unit increase in BMI

 A: 
Multiply the regression coefficient and the confidence limits on the
log-odds scale by $a$ and exponentiate to get the odds ratios and the
corresponding confidence interval for an increase of the predictor by
$a$. The $p$-value is not affected by scaling.

Alternatively, dividing the predictor by $a$ before fitting the model will scale the corresponding coefficient and confidence limits by $\times a$. Multiplying the predictor by $a$ will scale them by $/a$.
Let's confirm this by a quick example in R. We'll model the probability of being admitted into graduate school. The predictors are the GRE (Graduate Record Exam scores), the GPA (grade point average) and the prestige of the undergraduate institution. We're interested in the odds ratio and CI for an increase in $+10$ points of the GRE. We first fit the model using gre and multiply the confidence limits on the log-odds scale by $10$. For the second approach, I divide the original predictor by $10$ and refit the model using the scaled GRE. The coefficient and confidence limits are then for an increase of $10$ units of GRE.
# The data
dat <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
dat$rank <- factor(dat$rank)

#-----------------------------------------------------------------------
# Multiplication of coefficient and confidence limits
#-----------------------------------------------------------------------

# Logistic regression model
mod <- glm(admit ~ gre + gpa + rank, data = dat, family = "binomial")
# The p-value for "gre"
summary(mod)$coefficients["gre", ]

   Estimate  Std. Error     z value    Pr(>|z|) 
0.002264426 0.001093998 2.069863505 0.038465128

# Confidence interval on the log-odds scale for "gre"
confint(mod)["gre", ]
       2.5 %       97.5 % 
0.0001375921 0.0044358741

# Confidence interval for the odds ratio for "gre"
exp(confint(mod)["gre", ])
   2.5 %   97.5 % 
1.000138 1.004446 

# Confidence interval for the odds ratio for an increase by +10 in "gre"
exp(confint(mod)["gre", ]*10)
   2.5 %   97.5 % 
1.001377 1.045357

#-----------------------------------------------------------------------
# Scaling "gre"
#-----------------------------------------------------------------------

dat$gre_10 <- dat$gre/10

# The model
mod2 <- glm(admit ~ gre_10 + gpa + rank, data = dat, family = "binomial")

# The p-value for "gre_10"
summary(mod2)$coefficients["gre_10", ]

  Estimate Std. Error    z value   Pr(>|z|) 
0.02264426 0.01093998 2.06986350 0.03846513

# Confidence interval for the odds ratio for an increase by +10 in "gre"
exp(confint(mod2)["gre_10", ])
   2.5 %   97.5 % 
1.001377 1.045357

The $p$-value is identical between the two approaches ($0.0385$) and the confidence limits as well.
