Please Note that it is important to concentrate on the negative wording of the definitions for CR and DR if you are to grasp these concepts. Remember, these are in ASA 200.13 and A34-A46!
The audit risk model is AR = IR x CR x DR where:
AR = Audit risk (or acceptable audit risk in the case)
IR = Inherent risk
CR = Control risk
DR = Detection risk (or planned detection risk in the case)
One way the model can be restated is: DR = AR/(IR x CR)
This allows us to show that by holding AR and IR constant, there is an inverse relationship between CR and DR. That is to say an increase(decrease) in CR leads to a decrease (increase) in DR. To understand this, remember the negative wording of the definitions for these two elements of the model. (You may need to review ASA 200 to refresh your memory). How does this relate to the amount of audit testing? As DR decreases(increases) the amount of substantive testing increases(decreases). This means that the lower DR, the auditor wants the chance of NOT detecting an error to be smaller, hence the need to increase the substantive testing.
Here is a quantitative example (all % rounded to the nearest 1%). Assume AR is set at 10% and IR at 80% and will be held constant - this is normal because the factors that impact on these are unlikely to change as evidence is collected. If CR is higher, say 80% then the model shows a relatively low DR of 16% (.1/[.8 x .8]). However, if CR is lower, say 20% then the model shows a relatively high DR of 63% (.1/[.8 x .2]). Let me add that a relatively higher CR assessment reflects a higher risk that controls will not pick material misstatements; this means that controls are not very good and the auditor needs to gather (relatively more) evidence from substantive testing (hence the lower DR … a lower chance that substantive tests will not detect those same potential material misstatements. You might ponder whether as an auditor you would like to see stronger or weaker controls in place by management when IR is higher … now think about what you are likely to actually find when IR is higher?
This is an excellent example of where the professional terminology you are learning must be used in the way it is defined NOT how you might use it in everyday language use.