This question touches on both conceptual and statistical points, and I'd advise people in this situation to start with the conceptual part first.
There has been a popular phenomenon that has several ‘hallmark’ components, and are uniquely associated with sexual abuse survivors. I would like to verify that the components can stand as a construct, and that these components are indeed a unique phenomenon among sexual abuse survivors, more than other typologies.
I assume that the OP has a measure of some sort of symptom or trait that is unique to sexual abuse survivors. If you had omitted "unique", I would tend to guess it was something like post-traumatic stress disorder.
Whatever the case, in measuring psychological constructs like symptoms, we tend to reach for some sort of multiple-item scale. For example, the Patient Health Questionnaire-9 is a 0-27 point scale that measures depression. It has 9 questions. They cover affect (i.e. feelings, specifically sadness and inability to feel pleasure), cognitive (e.g. thinking you're a failure), and somatic (e.g. being fidgety, moving a lot more slowly than usual).
Most symptom scales undergo validation work. The developers will take some purposive approach to developing an item pool, i.e. a set of questions that get at the phenomenon they're trying to measure. The PHQ-9 developers were clinicians, and they just enumerated the symptoms of the DSM-III (the then-current version) - although the symptoms that get into the DSM to begin with get discussed in multiple rounds by psychiatrists relying on their practical experience, so it's not like they were generated at random. These authors had less of a knowledge base when they tried to develop a measure of quality of life for nursing home residents. (Disclosure: the first author was the wife of a late mentor.) They relied on intensive discussion with subject matter experts (policy makers and clinicians), plus resident interviews. Anyway, from there, you can select a set of items, maybe prune them or refine them using classical test theory or item response theory approaches.
When the OP says that they'd like to "verify that the components can stand as a construct," I would assume that they have not got a validated scale. It's unclear if the components they're referring to are a collection of individual questions, or (because they mentioned LPA, which typically refers to continuous indicators whose distribution is assumed to be Gaussian) if they have the sum scores of several separate tests, or if it's some collection of individual tests that produce continuous results (e.g. lab tests for hormones will return a continuous result), or something else.
How much do you know, theoretically?
I'll use "item" synonymously with a test question, e.g. one SAT question, one question on the PHQ-9. If there's already theoretical research that says that the construct in question is real, and there's already a validated scale, you can get by with doing no validation work. In fact, most randomized clinical trials do this. However, you can verify that the items in the scale hang together - that is, that they "can stand as a construct", by using confirmatory factor analysis.
You'd fit a structural equation model (SEM), positing that there's a unidimensional latent trait causing the item responses. (Assuming the scale was presented as a unidimensional construct.) If dealing with ordered categorical questions (like the PHQ-9 ones, where you're asking if someone was rarely bothered, sometimes bothered, often bothered, or almost always bothered by a symptom), you would want to use a weighted least squares estimator (as opposed to maximum likelihood) for technical reasons. You can then report a goodness of fit index, e.g. the Tucker Lewis Index, for which there are generally accepted cutoffs. Basically, it tells you how well the unidimensional model fits. If you're under the cutoff, that can indicate that your items are not hanging together well.
Alternatively, you could fit an item response theory (IRT) model, which is basically a generalized SEM model, and report the item discrimination/slope parameters. Items with low slopes (in education, I believe the preference is slopes > 1; I tend to repeat this in health services research but I have rarely seen slopes < 1) basically don't tell you much about the latent trait, and a lot of low slopes may implicate the construct or the measure of the construct. I don't believe there's a global test for good fit or not like the TLI, but I haven't done a lot of research in this.
If you were operating in a more exploratory mode, where you selected a bunch of items that you think may have something to do with one or more facets of the construct in question, you would probably want to do exploratory factor analysis (EFA). I believe that the 5-factor model of personality was developed using EFA; in that context, psychologists didn't have a great sense of what dimensions of personality even existed (apart from introversion/extraversion), so they had people rate themselves on a bunch of traits from the dictionary, then they ran EFA. They came out with, no surprise, 5 factors. In this context, you might not know what the facets or sub-factors of the overall construct are, so EFA could help.
I would consider the CFA or IRT approach to be testing for internal validity, specifically internal consistency. Note that Cronbach's Alpha is an older measure of internal consistency, but it has a number of downsides and I prefer not to use it. Normally, during test validation, researchers attempt to see how well their test correlates with or predicts a gold standard measure - for example, the PHQ-9 scores were validated against depression diagnoses made via clinical interviews. A gold standard may not be available - in the Kane et al paper, the person is ultimately the only reliable informant to their own quality of life, so there's no gold standard to validate against. I believe in psychometrics, this would be called showing criterion validity.
Discriminant validity
In psychometrics, discriminant validity means that the test you're interested in should not correlate with measures of different constructs. For example, a math test should not correlate too strongly with a verbal ability test. A depression symptom score should not correlate too strongly with, say, a measure of physical function. I realize this is not exactly what you are asking to do, but it is somewhat in the same vein.
In this case, you would probably need to define what other types of abuse might be related, and you would want to administer your test to samples of persons who have experienced sexual abuse and all the other comparison types. I assume the instrument in question produces a sum score that can be regarded as continuous. You could simply show the standardized mean differences on your instrument between the sexual abuse group and all the other groups. For example, Cohen's D is a pretty easy to calculate measure of the standardized mean difference - basically how many standard deviations do the means differ. 0.5 SD is usually regarded as a moderate difference, 0.2 as small.
What would latent class/profile analysis have told the OP?
I'll use LCA as a general synonym. Technically, I believe that LCA typically refers to binary, or at least to categorical items. LPA, as I mentioned, typically treats the items as continuous and distributed as Gaussian (but you can assume any distribution whose likelihood function you can write and that your software supports, e.g. you could simulate a mixture of Poisson distributions).
Depression can, practically, be treated as unidimensional. However, remember that I mentioned somatic items on the PHQ-9? One critique that some people make is that many older adults have a bunch of physical illnesses, and those might cause somatic symptoms and thus confound their responses to the PHQ-9. You might more generally wonder if there are some groups of patients with atypical responses to depression. You would not be the only one to wonder this; this is a systematic review of LCAs on depression scales. I haven't really dug into the findings. However, when you fit an LCA to symptom scales, you will almost surely find one high-symptom and one low-symptom group, plus maybe some intermediate groups. That's not very interesting. What's more interesting is the atypical groups. Some of the studies in that review found groups with high psychosomatic symptoms. A few found psychotic depression as their atyptical groups (but note that they may not have sampled from the general population and may have sampled from psychiatric treatment centers). Anyway, the atypical groups are the substantively interesting ones.
You may want to contrast LCA to exploratory factor analysis. What might have caused an atypical group of respondents? It could be that the construct is multidimensional, and one group of respondents has a higher level on one of the dimensions than the other. If the contention about older persons and somatic symptoms is correct, then if you ran an EFA on a sample, you might expect to be able to identify a group of somatic items. If you ran an LCA on that sample, you might expect an atypical group with a higher level of somatic symptoms. In this setup, EFA and LCA might provide similar information.
However, back to the question:
intend to use LPA to examine the determined typologies and assess whether the phenomenon (let's give it a name: construct A) is more pronounced in the group with SEXUAL violence than on the other groups.
LCA is not the best approach for the question explicitly identified in this statement. When I read the statement, I would tend to think more that construct A can be considered as unidimensional and continuous. So, you would just calculate and compare the means by group! Or if not unidimensional, you do that process for each identifiable dimension.
Under an LCA approach, you would be identifying qualitatively different typologies of construct A. You should treat them like an un-ordered categorical variable. You will probably find a group that's low in most symptoms, and a group that's high in most symptoms, so LCA can be used in this fashion. However, it is surprisingly not straightforward to regress these groups on an external independent variable (e.g. trauma type). An explanation is below this section.
If you object that you're not certain that your measure of construct A is really unidimensional (or close enough to unidimensional), then you would be better off trying to understand its dimensionality first, through methods like EFA.
LCA is hard!!
I would tend to recommend EFA over LCA to most applied statisticians. LCA has a lot of complex issues that can trip up applied analysts. For example, do you know how your statistical package handles estimates at the boundary of the parameter space? Specific to LPA with Gaussian indicators, do you know what your package's default treatment of the error terms' variance-covariance structure is, and what it implies? Do you know what might cause your model to fail to converge, and what are principled solutions to that? (Hint, many of them are related to the first issue.) Do you know that you should use numerous randomly selected start values, and why? Do you know that after an LCA, you don't have each observation's latent class? You really have a vector of latent class memberships. You thus can't directly take their latent class and do a multinomial regression on some other variable (which, by the way, is a little tricky to interpret to begin with). You could assign each observation to its most likely latent class, but this has measurement error that can bias your results. You can fit a latent class regression, which will simultaneously fit the LCA model and do the multinomial regression on the observed variables. Alternatively, there are approaches that correct for the measurement bias but are complex to explain. If you Google Jeroen Vermunt and his (and other authors') writings on three-step approaches, you will see what I mean. Stata definitely doesn't implement 3-step approaches, and I'm not aware of R packages that do; that would leave you with MPlus or Latent Gold, which are proprietary and quite rare softwares.
