It turns out, that the effect of violating sphericity is a loss of power ( i.e. an increased probability of a Type II error) and a test statistic ( F-ratio ) that simply cannot be compared to tabulated values of F-distribution. F-test becomes too liberal ( i.e. proportion of rejections of the null hypothesis is larger than alpha level when the null hypothesis is true.

Precise investigation of this subject is very involved, but fortunately Box et al wrote a paper about that: https://projecteuclid.org/download/pdf_1/euclid.aoms/1177728786

In short, the situation is as follows. First, let's say we have one factor repeated measurements design with S subjects and A experimental treatments In this case the effect of the independent variable is tested by computing F statistic, which is computed as the ratio of the mean square of effect by the mean square of the interaction between the subject factor and the independent variable.

In above article Box revealed, that when sphericity fails, the number of degrees of freedom $\upsilon_{1}$ of F ratio depends on a sphericity $\epsilon$ like so : $$ \upsilon_{1} = \epsilon(A-1) $$

Also Box introduced the sphericity index, which applies to population covariance matrix . If we call $\xi_{a,a}$ the entries of this AxA table, then the index is

$$ \epsilon = \frac{\left ( \sum_{a}^{ }\xi_{a,a} \right )^{2}}{\left ( A-1 \right )\sum_{a,a'}^{ }\xi_{a,a'}^{2}} $$

The Box index of sphericity is best understood in relation to the eigenvalues of a covariance matrix. Recall that covariance matrices belong to the class of positive semi-definite matrices and therefore always has positive of null eigenvalues. Thus, the sphericity conditionis equivalent to having all eigenvalues equal to a constant.

So, when sphericity is violated we should apply some correction for our F statistics, and most notable examples of this corrections are Greenhouse-Geisser and Huynh-Feldt, for example

Without any corrections your results will be biased and so unreliable. Hope this helps!

It turns out, that the effect of violating sphericity is a loss of power ( i.e. an increased probability of a Type II error) and a test statistic ( F-ratio ) that simply cannot be compared to tabulated values of F-distribution. F-test becomes too liberal ( i.e. proportion of rejections of the null hypothesis is larger than alpha level when the null hypothesis is true.

Precise investigation of this subject is very involved, but fortunately Box et al wrote a paper about that: https://projecteuclid.org/download/pdf_1/euclid.aoms/1177728786

In short, the situation is as follows. First, let's say we have one factor repeated measurements design with S subjects and A experimental treatments In this case the effect of the independent variable is tested by computing F statistic, which is computed as the ratio of the mean square of effect by the mean square of the interaction between the subject factor and the independent variable. When sphericity holds, this statistics have Fisher distribution with $\upsilon_{1}=A-1$ and $\upsilon_{2}=(A-1)(S-1)$ degrees of freedom.

In above article Box revealed, that when sphericity fails, the correct number of degrees of freedom becomes $\upsilon_{1}$ of F ratio depends on a sphericity $\epsilon$ like so : $$ \upsilon_{1} = \epsilon(A-1) $$ $$ \upsilon_{2} = \epsilon(A-1)(S-1) $$

Also Box introduced the sphericity index, which applies to population covariance matrix . If we call $\xi_{a,a}$ the entries of this AxA table, then the index is

$$ \epsilon = \frac{\left ( \sum_{a}^{ }\xi_{a,a} \right )^{2}}{\left ( A-1 \right )\sum_{a,a'}^{ }\xi_{a,a'}^{2}} $$

The Box index of sphericity is best understood in relation to the eigenvalues of a covariance matrix. Recall that covariance matrices belong to the class of positive semi-definite matrices and therefore always has positive of null eigenvalues. Thus, the sphericity conditionis equivalent to having all eigenvalues equal to a constant.

So, when sphericity is violated we should apply some correction for our F statistics, and most notable examples of this corrections are Greenhouse-Geisser and Huynh-Feldt, for example

Without any corrections your results will be biased and so unreliable. Hope this helps!