Hypothesis test between ordered categorical independant var and continuous dep. var Is there a standard hypothesis test for differences in the mean of a continuous dependent variable with respect to a single ordered categorical variable.  
By specifying that the independent variable is ordered, I also mean to impose the assumption that the dependent variable is monotonic in the independent variable.
 A: This sounds like anova-like model where there is hypothesized (before seeing the data) that there is a specific order among the means. Testing such hypothesis is called order-restricted inference, and there are multiple monographs on the topic (search this site). A simple model could be like a one-way anova, say with null hypothesis $\mu_A = \mu_B = \mu_c$ and alternative hypothesis
$$  \mu_A < \mu_B < \mu_C
$$
There is an R package restriktor let us try a simple simulated example:
library(help=restriktor)  
set.seed(7*11*13) # My public seed
mu_A <-10 ; mu_B <- 11 ; mu_C <- 12 # Alternative is true 
n <- 10
simdata <- data.frame(y=c(rnorm(n,mu_A,3), rnorm(n,mu_B,3), rnorm(n,mu_C,3)),
                      G=as.factor(c(rep("A",n), rep("B",n), rep("C",n))))
oneway.test(y ~ G, data=simdata) # Traditional anova to compare 
                                 # with results below

    One-way analysis of means (not assuming equal variances)

data:  y and G
F = 1.8923, num df = 2.000, denom df = 17.867, p-value = 0.1797

myConstraints <- 'GA < GB
                  GB < GC'  
mod0 <- lm(y ~ -1 + G, data=simdata) 
iht(mod0, constraints = myConstraints)

Restriktor: restricted hypothesis tests: 


Multiple R-squared reduced from 0.917 to 0.917 

Constraint matrix:
   GA GB GC    op rhs active
1: -1  1  0    >=   0     no
2:  0 -1  1    >=   0    yes


Overview of all available hypothesis tests:

Global test: H0: all parameters are restricted to be equal (==)
         vs. HA: at least one inequality restriction is strictly true (>)
       Test statistic: 4.3696,   p-value: 0.04507

Type A test: H0: all restrictions are equalities (==) 
         vs. HA: at least one inequality restriction is strictly true (>)
       Test statistic: 4.3696,   p-value: 0.04507

Type B test: H0: all restrictions hold in the population
         vs. HA: at least one restriction is violated
       Test statistic: 0.0834,   p-value: 0.7072

Type C test: H0: at least one restriction is false or active (==) 
         vs. HA: all restrictions are strictly true (>)
       Test statistic: -0.2889,   p-value: 0.6126

Note: Type C test is based on a t-distribution (one-sided), 
      all other tests are based on a mixture of F-distributions.

