Latent Class Analysis vs Rasch Analysis I was wondering if anybody could explain the core, philosophical/conceptual differences between Latent Class Analysis and Rasch Analysis.
I'm not necessarily referring to the exact mathematical calculations involved. I'm more interested in - why would one choose one over the other? Is there a core, fundamental difference or purpose between the two? Is one more suitable for certain types of data or research questions than others?
I'm primarily familiar with LCA but not Rasch, and most of the papers that I've read tend to go into slightly too much mathematical jargon that I simply can't focus on the arguments!
I was originally thinking about the two in the context of scale development but any other broader perspectives beyond scale development would be useful too.
 A: Latent class analysis is a latent mixture modeling technique that is interested in finding latent subgroups in multivariate categorical data. It quite naturally is a way to classify individuals, which might be useful in understanding how patterns of symptoms are related to different typologies. The continuous version of latent class analysis is called latent profile analysis.
Rasch analysis is concerned with the development of a measure for a single latent trait. The interest in Rasch modeling is to have items that fit the model so that sufficient statistics are produced. That is to say a measure developed using Rasch techniques would allow you to use the total score to summarize a persons total standing on the latent trait. This can be useful for comparing people for classification purposes. However, I don't think of the two techniques you mentioned as closely related. 
Please keep in mind I tried to keep this answer at a real surface level like you asked.
A: While I am not well versed in Latent Profile Analysis, I am familiar with Rasch. In response to bfoste 01's answer, assuming latent class analysis is described correctly, I feel Rasch does provide similar information as LPA, because in the dimensionaality output table, there is information provided about the presence of other constructs (other than that being measured). Rasch is a 1-parameter IRT method, where the data must fit the parameter of the model (which is determined by what the developer sets within the code: what responses are considered indicative of the construct being measured and to what degree it is representative of the construct- from less to more); while LPA, if I understand correctly, is an IRT method that allows the data to inform the model, meaning the parameters of the model can be "massaged" to fit the responses according how the respondents reacted (thus the model is made to fit the respondents). Imagine a ruler: would you rather the ruler keep "inches" the same for everyone (1-parameter: height) or would you like the ruler to respond to the people in your sample (so that if your sample was from a taller group, "inches" would be further apart than if you sampled another group of slightly smaller people). The difficulty with Rasch is that you have to be VERY knowledgeable about what you are trying to measure (which is difficult, given the latent quality of the constructs trying to be measured) AND be willing to go through multiple iterations. Most likely,you will need to go back and rewrite items to better reflect the construct. Item writing is very difficult and to do it well (so it creates a scale) means using respondents authentic language (which means focus groups, cognitive interviewing, and maybe discourse analysis). So it takes a mixed-method approach- which most ppl won't take the time/effort to do.
More information on Rasch can be found at www.rasch.org/rasch.htm for good beginner information. Also,a good primer book on Rasch is by Fox and Bond. The software for using Rasch is Winsteps (the easiest one for scale creation using 1-P...if you're looking to create a measure for a known multi-dimensional construct, there is Facets software. 
If LPA is different than that described, I would appreciate feedback on that as well. 
A: I agree with the comments made above by @Patti Lee and @bfoste01, though I believe this answer can add some clarity.
Both latent class analysis (LCA) and the Rasch model are models for categorical data (namely binary and ordinal data) that posit the existence of one or more (potentially related) latent variable(s). The core difference between the two methods is that LCA assumes the presence of a categorical latent variable(s). In contrast, the Rasch model assumes the presence of a continuous latent variable(s).
For example, say you have five binary (0/1) items designed to measure English proficiency to classify a student as either proficient or not proficient. Suppose you were to apply LCA to this data set. In that case, students could only be classified as proficient or not proficient (you would also have access to each student's probability of being either proficient or not proficient). If the Rasch model were to be applied, students would receive a score corresponding to their sum score (i.e., a score corresponding to 0, 1, 2, 3, 4, & 5). Then, if you want to use these scores to classify someone as proficient or not proficient, a cut point along 0 - 5 would need to be determined to reduce the six possible (continuous) scores into a binary classification.
The above example illustrates a scenario where either LCA or the Rasch model could be plausibly considered; however, not all applications of LCA are also appropriate for the Rasch model (and vice versa). For example, you discuss scale development, and LCA is typically not used for this purpose. However, the Rasch model is since psychometric scales are typically conceived to be continuous when the focus is to establish a scale to measure latent phenomena (though there are notable exceptions to this, for example, diagnostic classification model (DCM)s).
Finally, see Borsboom et al., 2016 for an accessible discussion of the differences between latent variable models for continuous (e.g., the Rasch model item response theory (IRT) models, and factor analysis (FA) models) and categorical (e.g., LCA,  latent profile analysis (LPA), and DCMs) traits.
References
Borsboom, D., Rhemtulla, M., Cramer, A. O., van der Maas, H. L., Scheffer, M., & Dolan, C. V. (2016). Kinds versus continua: A review of psychometric approaches to uncover the structure of psychiatric constructs. Psychological medicine, 46(8), 1567-1579.
