Linear Mixed Models and ANOVA 1) What is the difference between conducting a Linear Mixed Models and an ANOVA? 
2) In which circumstances do we conduct a Linear Mixed Models Analysis? 
 A: 
1) What is the difference between conducting a Linear Mixed Models and an ANOVA?

ANOVA models have the feature of at least one continuous outcome variable and one of more categorical covariates. Linear mixed models are a family of models that also have a continous outcome variable, one or more random effects and one or more fixed effects (hence the name mixed effects model or just mixed model).
There are sub-classes of ANOVA models that allow for repeated measures, a mixed ANOVA which has one within-subjects (categorical) covariate and at least one between-subjects (categorical) covariate, and repeated measures ANOVA which has at least two within-subjects (categorical) covariate and at least one between-subjects (categorical) covariate.

2) In which circumstances do we conduct a Linear Mixed Models Analysis?



*

*when we have a continuous outcome variable

*when data are clustered (for example, repeated observation on participants or students within classes)

*when we have sufficient number of clusters to enable estimation of the random effect (variance)

*when we are not interested in the "effects" of the clusters themselves.


Additionally, ANOVA cannot be used (though there may be a work-around), and mixed models offer a much better alternative, when


*

*we have missing data, or

*the experimental design is unbalanced, or

*we have multiple (cross-classified or nested) random effects, or

*we would like to allow the effect of covariates to differ among each level of a grouping variable (random coefficients or random slopes), or

*when we have an outcome variable that can't be plausibly considered as continous (such as count data and nominal data) - in which case we would use a generalised linear mixed model.



3) How do we obtain such a graph using the above model (Mixed Model or ANOVA) in SPSS to compare the "Low" and "High" condition of the product?

The figure appears to be a simple plot of means for 4 groups. Since it appears to be purely descriptive it isn't therefore something to be obtained from a model.
It appears to be typical of the type of data analysed with a two-way ANOVA - that is, a model with a continuous outcome variable, and two categorical covariates.
