The same random sites were used within the study area for quadrat sampling 35 years apart. The species richness and diversity has changed over time so I'm wondering what is the most statistically sound method of comparing these two results.
I am not sure if I understood your question correctly. If you want to compare the samples would be enough, a previous analysis a comparison between means.
Is there also any way of deciphering if for example an invasive species had a significant impact on the changes if the other abiotic parameters remained constant?
cualquier paquete estadístico te sirve, SPSS, SAS o los más específicos para estudios ecológicos. Creo que lo importante es el tratamiento o análisis que pretendes hacer con tus datos, qué piensas comparar y para qué? con esto en mente, revisas los supuestos estadísticos que deben cumplir tus datos para aplicar pruebas paramétricas o no paramétricas. Para las comparaciones de diversidad prueba con los modelos y varios índices, por ejemplo el alpha de Fisher es muy potente, pero lo ideal es calcular varios indicadores, Si piensas analizar patrones, entonces debes pensar en los métodos de clasificación y ordenación. Hay un trabajo clásico de Magurran 1985, sobre este tema.
maybe you can use linear models or generalized linear mixed models to test whether specific parameters have significant effects upon species richness and diversity. Using those models you are able to apply random factors to test whether the year of data collection has any effects upon the result. Or maby you could do a timeline analysis? I recommend using R.
4.Farrawy: Extending the linear model with R (https://github.com/clamm/R_dev/blob/master/Faraway%20-%20Extending%20the%20Linear%20Model%20with%20R.pdf)
I think you could try an ANOVA test only should see if (as Nestor says) your data meet the assumptions of normal distribution and considering the number of your "n" among others, because this is a test can give you lots of information and a program which serves a lot is "Statistica", give him some treatment to your data. I hope you find it useful comment.
Not sure exactly what parameters you want to look at for shifts over time, but if it involves community change you might like to have a look at the approach we took in attached.