If somebody wants to assess the phytodiversity loss due to plant invasions, is the use of diversity and richness indices of invaded and non invaded sites and their comparison enough for a preliminary assessment?
This can be used for preliminary assessment. You can compare the all data affecting/responsible for phytodiversity richness and loss. But you also keep in mind i.e. the effect of climate change on the weather pattern on both invaded and non-invaded sites. The invasions may have effect on the water level and other resources due to their Fast growth, Rapid reproduction, High dispersal ability etc. Hope it will be helpful for you. Good Luck.
An important (but commonly overlooked) principle is that a correlation does not imply causality. It would be normal to find that the abundance of introduced ('invasive') plants is negatively correlated with richness and diversity of indigenous plant species, regardless of whether the presence of the introduced plants is a symptom or a cause of phytodiversity loss. Therefore, whether your metric involves richness, diversity, keystone species, threatened species or anything else, it will not demonstrate that 'invasive' species have caused phytodiversity loss, only that they are correlated.
In some cases, it may be fairly clear that 'invasive' species are the cause (or at least a contributing factor); e.g. because they can be seen smothering other plants and inhibiting reproduction. Perhaps you already plan to do so, but I suggest you seek the actual effects of the introduced species on the indigenous species/community so that you can determine mechanisms and their magnitudes. You can't assume causality without information about causes.
However, in my experience in temperate Australia, introduced species are commonly just symptoms of underlying causes of phytodiversity loss, rather than being a cause in themselves.
For example, a human-induced disturbance of vegetation such as changed drainage may cause an ecosystem to decline or collapse, and opportunistic coloniser species move in to occupy niches being vacated by the species in decline. Of course, opportunistic colonisers feature prominently among plant species that are characterised as 'invasive', so it is not surprising to find a correlation between such species and declining phytodiversity, even when the species are merely responding to - not causing - the decline.
For that reason, I urge caution in using terms like 'invasive' or 'pest plant', because they are value-laden and - in cases like yours - tend to prejudge the effect of the species you're investigating. Terms like 'naturalised' or 'introduced' are value-neutral.
I should have added in my previous response that you can overcome the problem of correlation not implying causality if your experiment involves artificially adding the introduced species to sample sites where it was previously absent.
What does your experiment involve?
I can imagine an experiment with well-matched replicate sites, some of which are controls and others have one or more introduced species planted into them. Such an experiment is rare, I think, and it carries risks of permanent ecological damage to the study sites. However, it could demonstrate whether or not the planting leads to a statistically significant decline in particular species or biodiversity more generally, thereby overcoming the problem that correlation does not imply causality.
Your idea of using a measure of species richness or diversity is appropriate but there is also value in seeing which particular species or guilds of species decline or increase. The decline of some species (e.g. keystone species or threatened species) matters more than others, so it would be helpful to know how their abundance (and ideally, reproduction) change in response to the planting.
I suggest not using categories when recording data about abundance or cover. Where I come from, it is standard for people to record cover in categories such as '5-25% cover' (a Braun-Blanquet category). This means cover can change five-fold from 5% to 25% without any change being recorded, whereas if the cover remains constant at precisely 5%, it may be classified variously in the 5-25% category or the next-lowest category by different observers or on different occasions. I find it much better to record my best estimate of cover or population counts, ideally with a precision estimate.
Finally, you refer to a preliminary assessment. A short-term study might reveal that a certain introduced species causes a loss of pre-existent species even though a longer study may show that the effect wanes and the 'lost' species may eventually recover. Some species that are commonly regarded as invasive are quite opportunistic and explode in numbers initially, only to subside as the ecosystem adapts. I hope you get the opportunity to extend your preliminary assessment to a longer study.