Two different flows for air-fuel mixture are formed by two different variational amounts for changing the air-fuel ratio of the mixture. Response delay times for different exhaust gas flows are measured as time periods from a first time point at which the air-fuel ratio of the mixture is changed to a second time point at which an output of an oxygen sensor is changed. The oxygen sensor is provided in an exhaust pipe of an engine at a downstream side of an exhaust gas purifying catalyst. A difference between the two response delay times is compared with a predetermined threshold value to carry out the diagnosis for the catalyst deterioration.