To develop a no-reference video quality model, it is important to know how the perceived strengths of artifacts are related to their physical strengths and to the perceived annoyance. When more than one artifact is present, it is important to know whether and how its corresponding perceived strength depends on the presence of other artifacts and how perceived strengths combine to determine the overall annoyance. We study the characteristics of different types of artifacts commonly found in compressed videos. We create artifact signals predominantly perceived as blocky, blurry, ringing, and noisy and combine them in various proportions. Then, we perform two psychophysical experiments to independently measure the strength and overall annoyance of these artifact signals when presented alone or in combination. We analyze the data from these experiments and propose models for the overall annoyance based on combinations of the perceptual strengths of the individual artifact signals. We also test the interactions among different types of artifact signals. The results provide interesting information that may help the development of video quality models based on artifact measurements.