So in this experiment I have have 7 devices. The response of each device (call it Y) are each measured ~20 at a 4 different levels of an independent variable (we'll call X). The Y response is known depend on X, and this is clearly the case in the data.
What I am interested in finding out is if any of the devices have statistically different responses compared to the others for a given X. What is the appropriate ANOVA to run here? Should I run a different ANOVA at each level? I've heard of something called a block-design ANOVA; is that appropriate here, and if so could you explain it briefly or give me some good resources?
Thank you all for your help!
Based on the additional information in your Comment, here is the model for your experiment.
$$Y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk},$$
where $i = 1, \dots, 7$ devices, $j = 1, \dots, 4$ levels of 'X', and $k = 1, \dots, 20$ replications for each device-by-x combination, and $e_{ijk} \sim N(0, \sigma^2).$
This is a two-factor ANOVA with multiple replications per 'cell'. The ANOVA table will have rows for 'Device effect', 'x effect', 'Interaction' and 'Error' (or 'Residual'). There will be $F$-statistics for tests of significance of Device, x-factor, and interaction.
You will probably want to use software to make the ANOVA table. If there were $exactly$ 70 replications in each cell, this would be a 'balanced design'. If not, then you will need to use a 'general linear model' or 'GLM' procedure to get an approximate analysis, which should be fine as long as 'about 20' means something like 'between 15 and 25'.
You say you are particularly interested whether any one device performs significantly differently from the others for a particular level of factor x. If so, you should have a significant interaction effect. Then you can look at cell averages to get an idea which device(s) and which x-level(s) show interesting patterns and test a few 'ad hoc contrasts' to see if particular patterns are statistically significant.
If the 'interaction' effect is significant, that might interfere with the interpretation of 'main effects' for device and x-factor. For example, you might not be able to say a particular device has consistently higher or lower T-values than the others. Or that a particular level of the x-factor is best or worst.
You will have about $7 \times 4 \times 20 = 560$ observations, and hence as many 'residuals' from the model. You should look at a normal probability plot of the residuals to see if they are roughly normal. Also, if there are occasional far outliers among the residuals, you should consider whether the corresponding Y-measurements might have been faulty.
Balanced two-factor ANOVAs should be discussed in early chapters of any statistics text that covers experimental design. GLM for unbalanced designs may be discussed separately in a later chapter.
Note: Your question mentioned 'block' designs. Terminology about various kinds of block designs is not exactly standard from on text to another. Usually, block designs have only one observation per cell (or perhaps even strategically placed cells with no observations). In the simplest block designs there is no possibility of testing for interaction.