Multilevel regression model for nested data using "multilevel" and "lme4" R packages? Hypothesis 1
Independent variable = Thermal inertia (TI)
Dependent variable = Depth diameter ratio (d/D), Radii variation (RV) and Rim irregularity (RI).
Hypothesis 2
Independent variable = Thermal inertia (TI)
Dependent variable = Mantled Rim percentage (MRI)
For testing hypothesis 1, I have to test correlation of d/D, RV and RI with thermal inertia.
For 2nd hypothesis Correlation of MRP with thermal inertia.
Total craters = 138 (ID), Scenes (group) = 18. Repeated measurements were taken for each crater. Craters are nested within scenes, TI values are nested within scenes, and collected data (d/D, RV, RI, MRP) nested within scenes.
I have to compare TI both within and between scenes, also the statistical significance of each model result determined using a likelihood ratio test to attain p-values. Since I am new to R and statistics, if anyone could help me in understanding the design it will be of great help to me.
I have to look for p-values and results supporting hypothesis1 or 2?

 A: From what I can see of your data and the descriptions, you do not have multiple measures within ID. You have measured several variables, D_d, RI, RV, and MRP once for each ID.
Thus ID seems is the unit of measurements (that is, it unique to each row in your data).
However you do seem to have multiple measures within Group, and therefore a model with random intercepts for Group would seem to be appropriate. I would therefore suggest the following model as a starting point:
lmer(TI ~ D_d + RI + RV + MRP + (1 | Group), data = ... )

This will estimate fixed effects for D_d, RI, RV, and MRP, along with a variance for the random Group variable, which will account for the non-independence of measurements within each group. 
A: A couple of points:


*

*Mixed models are indeed used to account for correlations in your outcome variable, I guess TI within the levels of grouping/cluster variables, i.e., ID and Group in your case. Assuming that normal error terms would be adequate for TI, you could use a linear mixed model. For example, using function lmer() from package lme4, e.g.,
fm1 <- lmer(TI ~ RI + (1 | Group / ID), data = tisia)

If you are going also to load the lmerTest package, you will obtain a p-value for the association between TI and RI.

*Model fm1 above postulates that the correlation between any pair of measurements of TI within the same combination of levels of ID and Group is the same. If instead, you want to assume that the correlations within the same combination of ID and Group decay as the difference in RI values increases, then you could include the random slope for RI, i.e.,
fm2 <- lmer(TI ~ RI + (RI | Group / ID), data = tisia)


*You could compare the two models to see if this improves the fit using a likelihood ratio test implemented by the anova() function, i.e.,
anova(fm1, fm2)

