Are pregnancy tests effective a week before my period?

Yes: Modern pregnancy tests can give a positive test 3-5 days after conception and a week before the menses are due. A blood pregnancy test certainly can do this.

If positive: If the test result is positive, it is effective. However a negative result at less than 4-5 days after missed period does not rule out pregnancy. It is not advised to do home test before 4-5 days after a missed period. If the result is negative repeat the test 4-5 days later unless you get a period by then. Use first morning urine and follow the instructions for the test carefully.